You have a green cylinder, which is insulated, vented and about 30 years old. You have a power shower so the hot water, at least, is low pressure, but the cold might be high pressure.
You say the HW and the CH have to come on together so the cylinder is probably unpumped. You might have a 3 port valve but it sounds unlikely.
The immersion heater is probably in the top of the cylinder so it might heat about 40 litres, not enough for a bath. An immersion heater takes about one minute per litre. It has a thermostat inside so unless faulty should never be scalding.
I understand that after running the boiler for some hours, there is not much hot water in the cylinder.
Are the radiators getting "too hot to hold" on the top?
If the cylinder is upstairs, bleed the highest radiator in the house. Does water squirt out forcefully and without dying away after some seconds?