Interestingly the cold pressure always seems fine, and running the cold tap doesn't result in a filling sound coming from the cold cistern of the combination tank. I infer then that either there's a mains feed onto the bath cold tap, or there's a cold water tank in the loft. I'll have to go and have a look. Would either of these situations be a bit unusual?
I suppose with a pump there's always the risk of draining the hot cylinder faster than the feeder cistern can be replenished?