Contrary to what I've always understood, and scientists believed until recently, parrots and corvids use their tongues to modulate the "vowel" sounds 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 limited, though probably not completely removed (just as a human's would be limited, but not completely eliminated).
This is because birds' voices are entirely produced with a syrinx -- a much more versatile sound prodution organ than the mammalian vocal fold. The syrinx can, for instance, produce two separate pitches simultaneously, as well as a much broader range of waveforms. 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.
A bird's syrinx or similar apparatus, along with some method of changing the shape and length of the resonating cavities connected to it, are likely sources of "tongueless" speech incorporating approximatly the same range of sounds as human speech.