As such, I think the quickest a character can travel without teleporting in a single round is 15340ft, or 14160ft just in their turn. With a lot of luck for the Wild Magic shenanigans, Experimental Elixir of Swiftness and ally positioning, but it still gets you there!

Caveats

Admittedly, a lot of this is subject to the DM. As mentioned right at the beginning, I am using a specific interpretation where features that add to "speed" add to all base speeds (walking, flying, climbing, swimming, burrowing etc), whereas features that add to "walking speed" only add to the walking speed. Whether bonuses to "speed" are only meant to apply to walking speed is contentious and unclear, but I am going with the Crawford clarification that bonuses to speed apply to all speeds. Further discussion here.

It also depends on the DM allowing rolling for a Wild Magic Surge, which should be pretty reasonable but isn't a guarantee (and then the incredible luck getting the right Wild Magic Surge afterwards, though that isn't DM-dependent). More pressingly, it also depends on attuning to two artifacts both of which have 2 major beneficial properties to speed. Artifacts are only meant to be made by the gods so it's entirely up to the DM whether the required artifacts exist at all. After this, it's also unclear whether a single major beneficial property of speed is a unique magical effect and whether they stack – if the DM rules that they don't that reduces our artifact speed bonus by 30ft.

Lastly, we are getting a lot of mileage (lol) from the Tabaxi's Feline Agility racial feature. It is very questionable whether that still works when Wild Shaped or Shapechanged: you keep class and racial features, but only if the new form is physically capable of doing so. Unfortunately, Feline Agility has the barest minimum of fluff: "Your reflexes and agility allow you to move with a burst of speed." So it's unclear if it should work only on creatures that are sufficiently similar to Tabaxi (as it's a Tabaxi racial feature), creatures that are sufficiently feline (from the name Feline Agility) or any creatures that are sufficiently agile. This ruling is also very dependent on the DM, and losing that doubling will reduce the maximum speed quite significantly, so it's important to consider that if anyone is bold enough to try this in an actual game.

Result

Assuming all my assumptions hold, based on the above calculations the quickest a character can travel without teleporting in a single round is 15340ft, or 14160ft just on their turn. With a lot of luck for the Wild Magic shenanigans, Experimental Elixir of Swiftness and ally positioning, but it still gets you there!

• Race: Tabaxi, base 30ft walking speed, Feline Agility feature.
• Barbarian (Totem Warrior: Elk) 5: +10 speed from Fast Movement, +15 walking speed from Totem Spirit: Elk. (Needs to be raging for Elk Totem Spirit: do this as a bonus action on an earlier round, and make sure you keep your rage going until you can pull this off.)
• Druid 2: Wild Shape into Riding Horse (CR 1/4) for base 60ft walking speed. (As far as I can tell, this is the best way to boost your base speed and keep your other features. Shapechange is self only and we really don't want to go 17 Wizard; we need to do this other unholy class combination instead to get more speed boosts. Riding horse is the highest walking speed I can find, so it's good it's such a low CR that it can be Wild Shaped into from Druid 2. The Giant Eagle has a higher flying speed at 80, but as mentioned I am interpreting only increases to "speed" to work on flying speed. Wild Shape needs to be taken as an action on an earlier turn.)
• Fighter 2: Action Surge.
• Monk 6: +15 speed from Unarmored Movement.
• Sorcerer (Wild Magic) 1: Wild Magic Surge. (Requires casting a leveled spell and getting very lucky!)
• Wizard (Bladesinging) 2: Bladesong, +10 walking speed. (Use a bonus action to activate on an earlier turn.)
• +2 more levels in anything to get to lv20 to get Epic Boons.
• Mobile feat: +10 speed.
• Boon of Speed: +30 walking speed, bonus action dash once per short rest.
• Longstrider spell active: +10 speed. (Not concentration; cast it on yourself with your Wizard levels or have your friendly Wizard cast it on you.)
• Artifact 1 (1st attuned) with two major properties of speed: +20 walking speed.
• Artifact 2 (2nd attuned) with two major properties of speed: +20 walking speed.
• An ally Artificer (Alchemist) with the Level 3 feature Experimental Elixir and some luck gets you Swiftness, +10 walking speed.
• 2x ally Fighter (Battle Master) at level 3 for maneuvers, or anyone with the Martial Adept feat who has access to Maneuvering Attack: A friendly creature who can see or hear you can use its reaction to move up to half its speed. Having two is very important to allow us to use a reaction before our turn and a reaction after our turn, getting us two in one round.
• An ally Paladin (Oath of Glory) with the Level 7 feature Aura of Alacrity: +10 walking speed.
• An ally Wizard (Transmutation) with the level 6 feature Transmuter's Stone: +10 speed.
• Haste spell active: x2 speed, hasted action Dash. (Concentration, and 3rd level; have your friendly Wizard cast it on you.)
• Boots of Speed (3rd attunement): x2 walking speed. (Use a bonus action to activate this on a previous turn.)

All up, this gives us a base walking speed for the round of [60 (riding horse) + 10 (Barbarian Fast Movement) + 15 (Totem Spirit Elk) + 15 (Monk 6 Unarmored Movement) + 10 (Bladesong) + 10 (Mobile) + 30 (Boon of Speed) + 10 (Longstrider) + 20 (Artifact 1) + 20 (Artifact 2) + 10 (Experimental Elixir) + 10 (Aura of Alacrity) + 10 (Transmuter's Stone)] x 2 (Haste) x 2 (Boots of Speed) = 920ft.

As mentioned, on previous rounds we want to have used our action to Wild Shape into a riding horse, a bonus action to rage to activate Elk Totem Spirit, an action or bonus action to cast a leveled spell to activate Wild Magic Surge, a bonus action to trigger Bladesong, and a bonus action to trigger the Boots of Speed's doubling effect. We also need to have Longstrider and Haste active and be within 5ft of the ally Paladin with Aura of Alacrity. For the Wild Magic Surge (thanks @oneroi13 for spelling this out in this other answer!) we need the DM to have us roll after casting the leveled spell, then either get a 1 on a d20 or trigger it automatically with Tides of Chaos, then get a 01-02 on 1d100 on the Wild Magic Surge Table to trigger "Roll on this table at the start of each of your turns for the next minute, ignoring this result on subsequent rolls."

• Our ally Battle Master Fighter or whoever has Maneuvering Strike moves before us in initiative, hits a creature with a weapon attack, and triggers Maneuvering Strike from a place we can see or hear them, allowing us to use our reaction to move up to half our speed (410ft).
• At the start of the turn we pull this off roll 81-82 on 1d100 on the Wild Magic Surge Table to trigger "You can take one additional action immediately."
• Then we activate the Tabaxi feature Feline Agility: "Your reflexes and agility allow you to move with a burst of speed. When you move on your turn in combat, you can double your speed until the end of the turn. Once you use this trait, you can't use it again until you move 0 feet on one of your turns." This doubles our speed again to 1840ft.
• Then we Move, use our action to Dash, use our bonus action to Dash (from the Boon of Speed – or using a Ki point to use the Monk's Step of the Wind, which we also have), use our hasted action to Dash, use Action Surge and use that action to Dash, and use our extra action from Wild Magic Surge to Dash. That's 6 moves at 1840ft each, so 1840*6 = 11500ft.
• We end our turn within sight or hearing range of our second ally Battle Master Fighter, who luckily moves after us in initiative. We have our reaction back, so on their turn they hit a creature with a weapon attack, and trigger Maneuvering Strike from a place we can see or hear them, allowing us to use our reaction again to move up to half our speed. Unfortunately Feline Agility wore off at the end of our turn, so that's only another 410 ft.

In total that round, that's 11500 + 410*2 = 12320ft over 6 seconds, or 1400 miles per hour – just under Mach 2. So you can expect sonic booms as you move! Admittedly, the question was only about how fast you can move on your turn, so that's just 11500ft over 6 seconds, which is not much slower, and you can forgo the two Battle Masters.

@Ibenadar mentioned the 303 Monk in this other answer which can supposedly move at 16800ft per round while Wind Walking, and looking into it, it seems to assume that all increases to speed or walking speed stack on top of Wind Walk's base speed. I don't think that's reasonable, at the very least given how Wind Walk specifies flying speed while e.g. the Boon of Speed explicitly only boosts walking speed, so while it's supposedly faster than what I've given, I don't think it's actually attainable.

As such, I think the quickest a character can travel without teleporting in a single round is 12320ft, or 11500ft just in their turn. With a lot of luck for the Wild Magic shenanigans, Experimental Elixir of Swiftness and ally positioning, but it still gets you there!

Edit: as @David Coffron mentioned, it works to use Shapechange instead of Wild Shape: you still keep your class and racial features, and this character is capable of casting it from a Level 9 Spell Scroll with their Wizard levels (and with an Intelligence ability check of 19, but let's assume we make that). Shapechange is concentration, and is Target: self, so if we want Haste too we will need to have it cast on this character or use a Potion of Speed. As such we can remove the Druid levels and take more Monk, boosting the character's speed even higher.

• Race: Tabaxi, base 30ft walking speed, Feline Agility feature.
• Barbarian (Totem Warrior: Elk) 5: +10 speed from Fast Movement, +15 walking speed from Totem Spirit: Elk. (Needs to be raging for Elk Totem Spirit: do this as a bonus action on an earlier round, and make sure you keep your rage going until you can pull this turn off.)
• Fighter 2: Action Surge.
• Monk 10: +20 speed from Unarmored Movement.
• Sorcerer (Wild Magic) 1: Wild Magic Surge. (This helps us, but requires casting a leveled spell and getting very lucky! See below in the Setup section.)
• Wizard (Bladesinging) 2: Spellcasting (for the spell scroll) and Bladesong, +10 walking speed. (Use a bonus action to activate on an earlier turn.)
• This takes us to Level 20 to get Epic Boons.
• Mobile feat: +10 speed.
• Boon of Speed: +30 walking speed, bonus action dash once per short rest.
• Shapechange spell active: Shapechange into a quickling with base 120ft walking speed. Cast Shapechange with a Level 9 spell scroll using your 2 levels in Wizard, make the DC 19 Intelligence ability check, and make sure you maintain concentration.
• Longstrider spell active: +10 speed. (Not concentration; cast it on yourself with your Wizard levels or have your friendly Wizard cast it on you.)
• Artifact 1 (1st attuned) with two major properties of speed: +20 walking speed.
• Artifact 2 (2nd attuned) with two major properties of speed: +20 walking speed.
• An ally Artificer (Alchemist) with the Level 3 feature Experimental Elixir and a bit of luck gets you Swiftness, +10 walking speed.
• 2x ally Fighter (Battle Master) at level 3 for maneuvers, or anyone with the Martial Adept feat who has access to Maneuvering Attack: A friendly creature who can see or hear you can use its reaction to move up to half its speed. Having two is very important to allow us to use a reaction before our turn and a reaction after our turn, getting us two in one round.
• An ally Paladin (Oath of Glory) with the Level 7 feature Aura of Alacrity: +10 walking speed.
• An ally Wizard (Transmutation) with the level 6 feature Transmuter's Stone: +10 speed.
• Haste spell active: x2 speed, hasted action Dash. (Concentration, and 3rd level; have your friendly Wizard cast it on you.)
• Boots of Speed (3rd attunement): x2 walking speed. (Use a bonus action to activate this on a previous turn.)

All up, this gives us a base walking speed for the round of [120 (quickling) + 10 (Barbarian Fast Movement) + 15 (Totem Spirit Elk) + 20 (Monk 10 Unarmored Movement) + 10 (Bladesong) + 10 (Mobile) + 30 (Boon of Speed) + 10 (Longstrider) + 20 (Artifact 1) + 20 (Artifact 2) + 10 (Experimental Elixir) + 10 (Aura of Alacrity) + 10 (Transmuter's Stone)] x 2 (Haste) x 2 (Boots of Speed) = 1180ft.

As mentioned, on previous rounds we want to have used our action to cast Shapechange into a quickling, a bonus action to rage to activate Elk Totem Spirit, an action or bonus action to cast a leveled Sorcerer spell to activate Wild Magic Surge, a bonus action to trigger Bladesong, and a bonus action to trigger the Boots of Speed's doubling effect. We also need to have Longstrider and Haste active and be within 5ft of the ally Paladin with Aura of Alacrity. For the Wild Magic Surge (thanks @oneroi13 for spelling this out in this other answer!) we need the DM to cooperate, having us roll after casting the leveled spell, then either roll a 1 on a d20 to trigger a Wild Magic Surge or trigger it automatically with Tides of Chaos, then get a 01-02 on 1d100 on the Wild Magic Surge Table to trigger "Roll on this table at the start of each of your turns for the next minute, ignoring this result on subsequent rolls."

• Our ally Battle Master Fighter or whoever has Maneuvering Strike moves before us in initiative, hits a creature with a weapon attack, and triggers Maneuvering Strike from a place we can see or hear them, allowing us to use our reaction to move up to half our speed (590ft).
• At the start of the turn we pull this off roll 81-82 on 1d100 on the Wild Magic Surge Table to trigger "You can take one additional action immediately."
• Then we activate the Tabaxi feature Feline Agility: "Your reflexes and agility allow you to move with a burst of speed. When you move on your turn in combat, you can double your speed until the end of the turn. Once you use this trait, you can't use it again until you move 0 feet on one of your turns." This doubles our speed again to 2360ft.
• Then we Move, use our action to Dash, use our bonus action to Dash (from the Boon of Speed – or using a Ki point to use the Monk's Step of the Wind, which we also have), use our hasted action to Dash, use Action Surge and use that action to Dash, and use our extra action from Wild Magic Surge to Dash. That's 6 moves at 2360ft each, so 2360*6 = 14160ft.
• We end our turn within sight or hearing range of our second ally Battle Master Fighter, who luckily moves after us in initiative. We have our reaction back, so on their turn they hit a creature with a weapon attack, and trigger Maneuvering Strike from a place we can see or hear them, allowing us to use our reaction again to move up to half our speed. Unfortunately Feline Agility wore off at the end of our turn, so that's only another 590 ft.

In total that round, that's 14160 + 590*2 = 15340ft over 6 seconds, or 1400 miles per hour – just over Mach 2.3. So you can expect sonic booms as you move! Admittedly, the question was only about how fast you can move on your turn, so that's just 14160ft over 6 seconds, which is only slightly slower, and you can forgo the complex positioning with the two Battle Masters.

@Ibenadar mentioned the 303 Monk in this other answer which can supposedly move at 16800ft per round while Wind Walking, and looking into it, it seems to assume that all increases to speed or walking speed stack on top of Wind Walk's base speed. I don't think that's reasonable, at the very least given how Wind Walk specifies flying speed while e.g. the Boon of Speed explicitly only boosts walking speed, so while it's a bit faster than what I've given, I don't think it's actually attainable.

As such, I think the quickest a character can travel without teleporting in a single round is 15340ft, or 14160ft just in their turn. With a lot of luck for the Wild Magic shenanigans, Experimental Elixir of Swiftness and ally positioning, but it still gets you there!