Gollum would wither into nothing pretty much as soon as the Ring was destroyed (if he hadn't fallen to his own destruction). I imagine sort of like the Ringwraiths, as you said, who had "fizzled out" so to say. The only thing that kept their physical form in Middle-earth was the Ring and when that was gone, their lifeline was gone.
Bilbo ages rapidly after the ring's destruction, but still he's at an age that a physical body could sustain (at least for Hobbit standards). Gollum, being over 500 years old, he similar to a wraith (and was undergoing the "fading" thing, because the Ring doesn't increase one's lifespan it just continually stretches it. When the source is gone, Gollum's gone (and it would be instantaneous) because he was far beyond the years a hobbit body could sustain.
But I promise I'll try to return and not just be a spoil-sport, because I like the hypothetical question of Gollum's rehab post-Ring.
