Can you have HW on without CH, but not CH without HW?
Looking at that cylinder, you have a fairly ancient setup, I agree with Stem, and would have to question if it's ever worked properly. Gravity Primaries, (flow and return pipes from boiler to cylinder), need to be a minimum of 28mm diameter for a gravity system, 15mm is just not going to transfer enough heat, which may be your issue....
Basic premise is, boiler heats the water, which flows up to the cylinder by gravity. Some control was used in a few cases with a 'C' plan system, by use of a cylinder stat and motorised valve on the Primary return pipe, but this was quite basic in the grand scheme of things. If CH was required, the pump was simply switched on to move heated water from the boiler around the radiators.
What is the small copper cylindrical item in the last pic connected to?