Many moons ago, early DHW systems worked off a gravity-feed method. There was no pump and the water that heats the DHW just circulated around of its own accord.
When early CH appeared, the pump that was necessary to get water into the rads just did that - it was not used to heat the DHW cyclinder. If you have one that old, then the pump is probably knackered.
When the drive to improve efficiency in the system started to occur in the 80s, a diverter valve was inserted in the bolier water flow and both the DHW and the CH gets heated using the pump. If you've got one like that, it is probably the diverter valve that is knackered. It can't be the pump because you've still got DHW.