Good! I was going to wire it (have it wired!) from the switched side of
the illuminated mirror so that it came on when the mirror light was on - I
assume that if it's wired in directly to the lighting circuit it'll be
drawing current all the time?

Nope. It will draw almost no current when not in use. No need to run it from
a switched circuit.

Make sure that the shaver point contains an isolating transformer as
it is going into a bathroom.

Screwfix 61532. No mention of an isolating transformer. I assumed (not
always a good idea, I know) it would be ok as shaver sockets are fitted in
bathrooms...

It does have one. Clues that it has a transformer are that it has
multi-voltage and is the size of a double socket (but much deeper). Non
isolated ones are the size of a light switch and are 230V only.

Christian.