We can't say. A parrot can realistically mimic a can opener, an opera singer, cats and dogs and doorbells, Piano riffs and human speech. Not to mention the calls of other birds!
Parrots don't have big brains, their tongues are not as prehensile as ours; their lips and cheeks are not flexible, they have no teeth to brace a tongue against. We use all those to form sounds like "P", "G", "T", "SSHHH", etc: Parrots accomplish all those with vocal gymnastics in the throat, not the mouth, and we humans cannot tell the difference.
Which means a "standard" vocal tract already shared by most animals is probably enough.
The real changes are in the brain, in two parts. First is the ability to hear a sound and (somehow) translate that into whatever complex series of muscle controls would produce it. We hear a new word, and usually in a single try can produce that word. Not so for other sounds; for us.
But a parrot can hear a doorbell, and it's brain translates that into muscular controls that reproduce it so well it fools humans. (Yes, they have perfect pitch too, so even more neural equipment on the hearing side). That is about inputs; The second area of the brain is obviously that masterful control center of the output side: The vocal tract itself; timing and fine control over the shape of the throat and tongue, air pressures, pitch, tone and volume required to reproduce sounds with such precision and accuracy, and speed of changes.
Exactly what changes are required in the brain are impossible to describe; we truly do not know that much. But you can be certain there are genes to lay down this neural machinery: The proof is in the pudding, Parrots can speak perfect Chinese, Russian, and English. Squirrels suck at Chinese!
Which means, Logan, give the animals a voice box with muscular control, and you are set. The neural machinery can fit in a parrot's cute little head, the necessary nerves and muscles can fit in their little throat.