Why does my shower keep alternating between freezing and boiling by itself?
We spoke to a plumber. Speaking off-record, he thought the problem might not be related to the new tap but may be incidental.

We now think its because of the pressure in our boiler. About a week after I posted this question, the pressure reached zero. We didn't realise that we had low pressure until it briefly stopped working! After re-pressurizing it, the shower temperature seems to behave reasonably. We still have an over-sensitive tap, but at least the shower temperature no longer changes by itself!

If you have this problem as well, check that the pressure in your boiler is within the expected range. For us, that's between 1 and 1.5 millibars.