I know I'm super late to this post, but here goes nothing.
My fastest speed build is as follows (using only what is listed in RAW, as well as 1 factor of presumed RAI, being that Dashing multiple times doubles speed each time, the reason for that assumption is that 5e is awful with wording sometime, and it seems to me that movement and speed are interchangeable, as some spells see them as such).
Keep in mind, that the wording of the Dash action states that it includes modifiers to your speed, which would include other iterations of the Dash action. This build also requires two friends, one magic item, and 2 total turns to do.
Tabaxi (30 base speed)
Mobile feat (+10 base speed, now total of 40)
Monk 18 (+30 base speed now total of 70, subclass is up to player choice)
Fighter 2
Longstrider spell from a friend (+10 base speed, now total of 80)
Haste spell from a friend (for double speed and extra Dash action)
Boots of Speed (for double speed)
Turn 1: Friends cast Haste and Longstrider on you, you use your Bonus Action to activate the Boots of Speed (base speed on your own of 70, now has +10 from Longstrider and is doubled from both Haste and the Boots for total of 320 ([70+10] * 2 * 2))
Turn 2 Your current base speed is 320 (see the math above). Dash Action (640), Step of the Wind for Bonus Action Dash (1280), Action Surge Dash (2560), Feline Agility (5120), Haste Action Dash (10240).
This build already requires two other people's effects, or I would've included a Transmuter's Stone. The goal in this is to maintain fairness, because there is no guarantee that you would have a Transmutation Wizard, but it is highly likely to have allies who can cast Haste and Longstrider. The math (if this was maintained over time, which it cannot be, due to spell length and such), is as follows: 10240 * 10 (feet per minute) = 102400 * 60 (feet per hour) = 6144000/5280 (miles per hour) = 1163.63636 miles per hour.
Sorry if this is disorganized, not the best at that kind of thing.