Close to the boiler, or close to the cylinder, there will probably be a pump (probably dark red) and a 3-port valve (brass, with three large pipes going into it in a T shape, and with a rectangular box, probably metallic grey but might be plastic. The fault is probably in the 3-port valve head, which contains a switch, or in the thermostat which is strapped to the side of the cylinder.
Verify first that the immersion heater is definitely turned off. It might have its own circuit breaker of fuse in the "fusebox" that you can turn off.
Then turn the cylinder stat up and down. You should hear it click on and off. You might hear a whir from the motor in the rectangular box on the 3-port valve.
If I am right, the boiler does not have to be opened up (it is quite tricky on your model) and the fault can be fixed by changing the motor head. Unless you are fond of plumbing and electrical work, you will need a heating engineer. If you are really lucky it will be the cylinder stat but they rarely go wrong.
Until fixed, use the timer to turn the boiler on and off. It should be able to heat the cylinder from cold in about half an hour, and the cylinder will hold about a bathfull.
You may find if difficult to get that model of boiler serviced or repaired, so ask around for recommendations of a good local engineer. Avoid yellow pages, and especially avoid advertising recommendation websites.