chris.fellows wrote:
I live in a 1st floor flat and have an overflow pipe which is slowly
dripping outside. The pipe is white, about 1 inch diameter and exits
the building at floor height.

I suspect that it may be my immersion heater or bath. Could someone
help me out?

It's coming from the lavatory cistern, cold water tank, or central
heating header tank. First of all get the lid off the cistern, and
see whether the water gets up to the overflow pipe after a period
of non-use.