Jelly, 16 bytes

e€ØcT⁸ṫ“shm”;ⱮḢȯ 

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 ⁸ṫ Remove a number of letters from the start of the word equal to T each truthy index of € for each letter, e Øc  is it a vowel?   “shm”;Ɱ Prepend "shm" to each,   Ḣ  take the first (or 0 if empty),  ȯ and replace a falsy result with the original word. 

Jelly, 16 bytes

e€Øco\x@ṭ⁸¹?“shm 

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Jelly, 16 12 bytes

œlØḄṭ⁸¹?“shm 

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-4 thanks to ovs remembering there's a strip builtin

œlØḄ Strip all leading consonants. ṭ “shm Prepend "shm" ¹? if there's anything left, otherwise  ⁸ give the original word. 

Jelly, 16 bytes

e€ØcT⁸ṫ“shm”;ⱮḢȯ 

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 ⁸ṫ Remove a number of letters from the start of the word equal to T each truthy index of € for each letter, e Øc is it a vowel? “shm”;Ɱ Prepend "shm" to each, Ḣ take the first (or 0 if empty), ȯ and replace a falsy result with the original word. 

Jelly, 16 bytes

e€ØcT⁸ṫ“shm”;ⱮḢȯ 

Try it online!

 ⁸ṫ Remove a number of letters from the start of the word equal to T each truthy index of € for each letter, e Øc is it a vowel? “shm”;Ɱ Prepend "shm" to each, Ḣ take the first (or 0 if empty), ȯ and replace a falsy result with the original word. 

Jelly, 16 bytes

e€Øco\x@ṭ⁸¹?“shm 

Try it online!

Jelly, 16 bytes

e€ØcT⁸ṫ“shm”;ⱮḢȯ 

Try it online!

Jelly, 16 bytes

e€ØcT⁸ṫ“shm”;ⱮḢȯ 

Try it online!

 ⁸ṫ Remove a number of letters from the start of the word equal to T each truthy index of € for each letter, e Øc is it a vowel? “shm”;Ɱ Prepend "shm" to each, Ḣ take the first (or 0 if empty), ȯ and replace a falsy result with the original word.