To answer my own question, you can exchange a UK licence for a Taiwanese one by popping into any Driving Licence Office in Taipei (and other cities I assume). You just need:
- Your passport
- Your A(P)RC (unsure if there are validity requirements for an ARC)
- Your British driving licence
- Three 1" passport-style photos
- A 'share your driving licence’ code generated by gov.uk to give the TW gov access to your full driving licence record (this avoids the need to get your driving licence notarised, I believe).
You DO need to undergo a health check. This takes 5 minutes and costs NT$300. I recommend going to this driving licence office
because you can walk over to Pojen Hospital for the health check. It’s a 2 minute walk away. At the hospital, they check your eyesight, height, weight, grip strength, hearing, and ask if you have any ongoing medical conditions. I suppose you can do this before you go to apply for your licence, but they told me I can just bring the results when I go back to pick up my TW licence.
After I applied for the driving licence, they said they’ll call when it’s ready for pick up. They said expect to wait 1–2 weeks. When I go back, they said to bring the health check and also provide them again with my passport, A(P)RC, 1" photos, and UK driving licence. I believe it’s $200. (I read that online but forgot to double check.)