A wiggle of the gear stick seems to bring them on, so I presume the fault is most likely the reverse switch? TIA

Yup switch. Easy to change, lives just to the right of the selector mech on the top of the gearbox, looking from front of the car. Ideally need to take the battery and battery tray out to get decent access though. Check PAS fluid at the same time, two birds with one stone. Or a Keith Richards service