Most likely you have Y-plan, possibly W-plan as it's all fairly old.
I meant gravity through the cylinder coil, olde-worlde system, which you haven't got. That's why I said primary. With your non-pressurised HW cylinder the flow to the taps (secondary side) must be by gravity.
If you change the boiler you could just do a direct swap for another heat-only, that would be simplest. But if the current one is working OK no reason to do that, and it wouldn't cure your cold rads problem. IMO the way to go is a proper power flush, or take the rads off one at a time, take them outside and flush with a hose.