Birds don't use their tongues in any way when they mimic sounds (songs of other birds, environmental sounds, animal calls, and even human speech). If a parrot or mynah were to lose its tongue, its ability to "talk" would be in no way lessened (though its ability to eat might be).
This is because birds' voices are entirely produced in their syrinx -- with no significant modulation (other than resonant amplification of certain pitches) from the rest of their respiratory system. This is why, for instance, a parrot can just as easily reproduce the sound of a film camera's shutter and motor drive or a microwave oven's "done" alarm as it can a cat's meow, dog's bark, or "Polly want a cracker?"
This vocal apparatus is common to all birds; the limitations on their ability to mimic is in their brain -- sparrows aren't primarily mimics, so they tend to learn one sparrow song and sing that for their entire lives. Crows, mynahs, magpies, parrots and parakeets, on the other hand, are mimics, and all are capable of learning, to some extent, to replicate any sound they hear frequently.