As far as I can tell the law relates to being employed in Taiwan and drawing salary here. You are not permitted to interact with the local economy in such a manner that you draw payment for your services or might theoretically draw payment for those services, formally or informally.
It goes without saying that most tourists and all business travellers are technically still employed in their home countries, and in the latter case are ostensibly working for their employer while in Taiwan. IIRC there is (or was) a different kind of landing visa granted for business travel, but in practice everyone came and went on a tourist visa. What the law is in Taiwan, and how it’s actually implemented, tend to be two different things, not least because the Law hasn’t caught up with the fact that people do move around to do their work (digital nomading in Taiwan is a very awkward grey area).
The tax issue is separate and clear enough: you are liable for tax only in the UK, because you are ordinarily resident there.
Realistically, is anybody even going to know anyway, if you’re locked in a room with a laptop?