I need to move a shaver point from one wall to another and install an
illuminated mirror above it in the bathroom. I assume Part P applies.

Definitely. Consider using a small mirror heater pad as well. Great things
and not that expensive. Switch it with the mirror light.

http://www.discountedheating.co.uk/s...Demisters.html

The present shaver point has a light built in and is wired straight off
the lighting circuit. Would this be acceptable in the new location or
should it be on a fused spur?

It is acceptable straight off the circuit.

If so, can the FCU be in the bathroom close up to the ceiling or must
it be outside?

You don't need one.

Christian.