Taiwan does not need to start a nuclear program. Taiwan has previously completed everything except final assembly of a nuke, so they have the capability in principle. There are a number of countries that have never built nukes but that are believed to possess all of necessary components to rapidly assemble nukes should they feel it necessary, it is a gray area in nuclear non-proliferation. For international accounting purposes, a disassembled warhead sitting on a shelf somewhere doesn't count as a nuke, they only count systems.
Invasions are unavoidably telegraphed months ahead of time, which is more than enough time to assemble a nuke if you have already have the components.
TW is still thoroughly infiltrated by PRC intelligence, if there's credible hint they're moving from nuclear latency/threshold to actual nuclearization (PRC redline), the relevant facilities would be destroyed - no need for PLA boots on ground or telescoped build up, every inch of TW is within 10 minute strike of mainland based ordnances. TW nuclear turnkey potential isn't as fast as PLA rockets/bunker busters.
Unmentioned aspect of US axing TW nuclear program in late 80s, was CSIST / INER (Taiwan nuclear program) was already monitored by PRC intelligence and post Nixon normalization US intel cooperated with PRC intel to shut the program down. Unwritten between the line is PRC, having closely monitored TW program would have probably nuked TW first if US didn't compel TW to end the program. US Inspection / IAEA went in the dismantle the program, no doubt verified by PRC intelligence, it's unlikely TW can rebuild to nuclear turnkey faster than PRC can mount an invasion. And realistically TW can't fend off a PRC invasion without US involvement, and if US involves to assist TW nuclearizing then TW going to get nuked, and US + PRC will stare down MAD.
I have a feeling that China is very prepared to invade with extreme haste and their intelligence agencies would not let something like this go unnoticed.
The sheer logistics of a large scale mobilization means that its obvious weeks and even months in advance. Moving troops, equipment and ammunition to staging areas etc. Even just preparing transports shows up.
Think back to the weeks before the Ukrainian invasion. It's kinda hard to hide it.
As far as I know, China has not been hiding this preparation. They are actively preparing to invade Taiwan, at least for the sake of appearances so that they can be seen doing so.
Invasions are unavoidably telegraphed months ahead of time, which is more than enough time to assemble a nuke if you have already have the components.