Shower is on the same level as the boiler and is a direct feed from
the hot water (ie not electric).
There is no cold water tank in the loft - is it possible that a) I'm
blind, b) I need a stronger torch, or c) there's one to service
several of the flats?

Or (d) you have no tank at all. Most flats, particularly new ones, will have
a combi boiler to heat the water. This requires no tanks or cylinders at
all. Everything runs off the mains. However, if the water supply is a bit
flaky, or all the internal pipework is off a 15mm run snaking around the
house, then you can easily get problems with outlets affecting each other.

The solution depends on whether the supply is the problem, or the poorly
designed pipework inside.

Christian.