In reality, you do not get to pause after an American fleet gets wiped and calmly decide what to do over tea...
But neither are movements done in a berserker fashion. If an American fleet is sunk, the American reaction is not going to be knee-jerk. And certainly a
nuclear response isn't going to be decided by a battlefield commander.
A carrier fleet is destroyed. The losing side has to figure out what to do. Resources are not moved instantly. It takes time. Do they reinforce? Regroup? Withdraw? All that takes time to decide. Ask anyone who has been on the front line. Macro-decisions are not made on the split-second.
You're describing a single battle played out by field colonels. (Or with the Navy, captains and rear admirals. They don't use commodores anymore, do they?) Nuclear war is decided by generals tens of thousands of miles away.
I can tell you how it will go down.
An American fleet is sunk by the Chinese.
The local commander decides how to respond using the current parameters he is given.
The Fleet Admiral makes the decision on how to respond with the resources under his command.
Back home, the Admiralty reports to the President the loss of the Fleet. The President then decides whether to change the Rules of Engagement, comes up with a new plan, orders an escalation, or whatever.
Meanwhile, the clock is ticking.
In all likelihood, the US would withdraw from Taiwan before using a nuke, presuming the war stays over there and China does not attack mainland assets (including cyberwarfare or space warfare).
That's an "if", of course. But that was the scenario given.
If we want to talk about an expansion of the battlefield, that is a different conversation.