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.