Thanks to all for the info so far.

I've had a closer look and there are 3 pipes going in to the zone valve.

I assume one comes from the boiler, one goes to the HW cylinder (which
does have a cylinder thermostat) and one goes to the pump. There is a
smaller pipe that comes back from lower level on the cylinder and joins
the pipe going to the pump.

The electrical box on top of the zone valve only has one screw on it -
high up on the opposite end to where the cable goes in and the little
lever sticks out. I haven't tried unscrewing it yet - is this wise with
the power still on? It doesn't look like this screw would release the
electrical gubbins from the valve, but I suppose it might give access to
screws that do.

I have fiddled with the little lever that sticks out. It was in the
central position, but if I move it to one side or the other, it seems
content to stay where I move it to.

I have also tried putting the lever to the central position, then
switching on just the heating - the lever doesn't move. Same thing if I
try putting on just the water.

Does any of this give a clearer idea of exactly where the problem lies?