Once again, completely ignoring that Taiwan would not be defending itself alone. Australia, Japan, South Korea, as well as the US, and most of Europe and the UK Commonwealth would also be jumping in to help defend it (and a good chance Vietnam and Philippines and even India as well). What, you honestly think it would be just the PRC and ROC?
As I said, you really do not think things like this through very well, do you?
Let me give you the reality.
Ok.
0. In our reality, Russia already said, that Taiwan is a part of People's Republic of China.
1. US and other international intelligence agencies detect the buildup of Chinese forces weeks in advance and the readying of their naval forces to put to sea towards Taiwan.
1. But they don't detect, that Russia also readying their forces and Emercom to fight a nuclear war, and sell some nuclear weapons to China.
2. UN and the International Community gives stern warnings that such will not be tolerated.
2. All atempts to make such warning in UN are blocked by Russia and China. Both Russia and China are declaring, that any possible conflict in Taiwan is the 'inner business of PRC' and no intervention will be tolerated.
3. As task force puts out to sea, US is already mobilizing all local assets to the region. Likely placing some US Naval assets between Taiwan and China, while most remain to the east.
3. CIA , finally, discover Russian preparation for a preventive nuclear strike. Biden have to decide
a) to start his own preparations to the preemptive nuclear strike, which can provoke Russian to further escalation, or
b) to step back and cancel preparations for a conventional war, or
c) to ignore Russian preparations (because 'the Russians won't attack first - it is unthinkable') and be still vulnerable.
Now either:
4. China backs down and goes home.
4b) the USA backs down and goes home, Taiwan lost hope and surrender.
Or:
4. China launches first attack. At which point it becomes an "open fire" zone. And all assets of Australia, US, England, and any other nations go to work on any Chinese forces they can.
4a) there is uncontrollable escalation, in which Russia, likely, attack first with a significant, but not overwhelming superiority. After the first nuclear exchange Russia had lost ten million people, China - fifty million people , the USA - one hundred million people. Taiwan either surrendered or scorched down.
4c) there is a 'sitting duck scenario', Russia eliminate 90% of the US nuclear arsenal and force Baiden to surrender (may be, even, unconditional surrender).
6. China gets a single wave into place (maybe). International cordon is placed around the island, destroying any PRC ships or aircraft that come close. Meanwhile, the 3rd Marine Division (Okinawa) as well as any other Marine assets (roughly 2 Regiments) as well as all other assets are mobilized to Taiwan to root out any Chinese that have managed to get ashore.
7. A radical change in the UN.
And how radical, will depend on what China does. I bet at a minimum, official recognition of Taiwan again. And that is the best of outcomes I could see for China. Absolute worst? Revocation of "Official Recognition and Status" as a member of the UN, and the seat and Veto Power being given back to Taiwan.
Your scenario is absolutely unacceptable for Russia, therefore, Russia won't allow it by any cost (including the cost of the nuclear war).