Thanks for the replies. I think I will renew just to be safe but still can’t see where we would have an issue if travelling on the Taiwanese passport.

I don’t have a good understanding of how the passport details on the booking tie up with the passport used at immigration/passport control. I have always assumed that they are separate so you are always free to use the most appropriate one when going through immigration even if you used another one for your flight booking. Although I understand that airlines will check that your passport is valid to enter the destination country. On leaving TPE I’d expect that simply showing them that she has a UK passport would be enough to satisfy the airline.

I also checked TIMATICS and neither the Netherlands nor UAE appear to have an issue with transits on passports with less than 6 months validity.