I am fitting a power shower in my bathroom, and need a 3A switched supply.
Is it ok to spur off my imersion heater supply with a 3A switched fuse?
This
is the nearest available supply and has its own supply from the consumer
unit.

Very probably. The immersion is probably a 3kW @ 240V, taking 12.5A. No
diversity would be allowed for this. However, I doubt the pump takes more
than 2.5A (probably more like 1A continuous), so it should be fine for a 15A
circuit. Obviously there is no issue for a 16A circuit either.

Christian.