with a copper cylinder, I expect it to be on a vented system, so there is a good chance that you have a sediment blockage. If you bleed the highest radiator in the house (preferably higher than the cylinder) does water squirt out forcefully and without dying away?
I am fairly confident that you will have a circulating pump, which should start when you turn the water heating on, to transfer heat from the thermal store to the coil in the HW cylinder. This is the other thing likely to be at fault. If you turn on the HW, this pump would get very hot if water is not circulating through it, which I think is likely.
Neither of these probable causes is a DIY fix unless you are fond of plumbing.
If it does turn out to be a sediment blockage, then as well as cutting out and replacing the blocked piece of pipe, ask the plumber to fit a System Filter, which will cost about £100 plus fitting, so is best done at the same time. If, as I think, you have a small feed and expansion tank, ask him to point it out to you, and you can paint the name on the side with a bit of white paint. If you are very lucky, the fault might be that the ballcock is old and jammed, which will be the easiest and cheapest fix.
As the causes I have in mind do not need work to be done on the thermal store or the boiler, any experienced and recommended local plumber can fix them. You will need a qualified person if it turns out to be your boiler, store or controls, though. You might prefer to go straight to your boiler maintainer, who probably has a higher charge rate.