Paul, my next step would be to cut the wires at some point where you can get to them easily. Then use a test meter on the ends that come from the tap (bare the wire ends first) Check for open circuit when the tap is closed and continuity with the tap open - i.e. after it has clicked. If this is OK, then the mocroswitch in the tap is functioning normally, and the fault is either in the 12v. feed to the switch or in the wire to the pump.
Put the test meter onto the voltage dc setting and check the wiring which goes to the pump. You should see 12v, (or thereabouts)> if this is not present, then you should check back right to the pump on switch which you should have - probably with a green light - on the caravan control panel. But since the rest of your taps are working, i am more inclinded to suspect the microswitch.
It is possible for the switch to click and still not work. On some british vans the switch is fastened to the tap with a small cradle and one of the wires may have come off. With mixer taps, it is possible for the wiring to have been installed incorrectly and become tight and even pull away from the switch.
You can take the handle off the tap bu prising out the red/blue disc which you propbaly have in the handle and then removing the screw underneath. Pull the handle off gently and have a good look at the switch. Unfortunately i have not found a source of new microswitches for the Reich mixer taps, but someone else may be able to advise. it seems over the top to have to buy a whole new tap just to replace the switch.
You can re-connect the wires using either crimp on connectors or "chocolate block" type - the circuit will be 12v. d.c. and ok to work on but avoid touching ends of wires together..
There will probably be only one fuse for all taps and the pump and since your other taps are ok I think the fuse is likely to be ok too.
Good luck
If any other reader can advise - particularly about spare micro switches please don't hesitate !