I doubt it. Seems to me that mainland China has accepted Taiwan's existence not only because of US support, but because Taiwan has (largely) stayed quiet. That is, they have made fewer noises about official independence than they could have and they have represented no military threat to the mainland. The simple development, let alone the actual detonation, of a nuclear weapon by Taiwan would change that. Relations would heat up very quickly and there would probably be a shooting war. In fact, I think the mainland would take that step well before Taiwan had a chance to finish a prototype, let alone explode one.
The only question is the role of the US in this. I can't see the US encouraging Taiwan to develop a bomb, or supporting them if they did it anyways, simply because it would be so destabilizing. And without US backing, they'd probably get overrun.