Plumber suggest doing something(?) from pipework from radiator in bedroom which is next to the bathroom
It sounds like your plumber wants to connect the cylinder heating coil as if it was another radiator. It will work but think carefully. You will only be able to get hot water if the heating is on. Do you really want to do this in the middle of a heat wave?
I see no reason why you shouldn't have a fully pumped system with the pipes you already have. The only difficult bit will be getting a cable to the cylinder thermostat which you say you don't have but will need. Also the feed/expansion pipe might need re-routing.
Meanwhile, back to the original problem. You can get HW to work on its own. You can get CH to work too but every time it switches off the boiler erupts. Here's a few tests you can do that might help. You will need to identify the heating coil pipes on your cylinder. They will be low down, one above the other on the same side (I hope).
1) Starting with a cold boiler, switch on HW only. Which boiler pipe gets hot first? Which heating coil pipe gets hot first. Please tell us it's the top one!
2) Starting again with a cold boiler, switch on HW and CH together. Make sure your room thermostat is turned up so that CH comes on. The pump should start. Which boiler pipe gets hot first. Which cylinder heating pipe? If my theory is right
they will not be the same ones!
PS: It's not ideal to have a gravity system with long runs of horizontal pipe but it can work. We inherited one when we moved house. It was infernally noisy when HW started up but it worked for years.