I'd probably renew the passport just to keep things simple. That said, I do know many dual US/Taiwan nationals who always enter/leave Taiwan on their Taiwan passports, so there should be no problem on the Taiwan (government) side. I'm not familiar with the UK side of things, so someone with more experience can hopefully chime in.

The airline that you take departing TPE back to the UK will presumably check that requirements for entry into the UK (e.g., visa) are satisfied if you choose to use the Taiwanese passport, so make sure that everything is OK on that end. The UK does offer visa waiver to Taiwan passport holders, but I noticed that the wikipedia article on UK visa policy goes out of its way to mention that the waiver is only offered to Taiwan passport holders with the household registration number. Presumably that's how the Taiwan passport was obtained in the first place, but I wanted to note that specifically.

My advice would be different if you had an older son.