Hi,
a bad earth would affect the other tail lamps as it's a shared earth connection on the rear lamps. The gearbox reverse light switch switches the feed to the lamps, so I would check the Green wire to the switch has 12volts going to it(if not you have a bad connection in the wiring harness on the green wire) and the same voltage coming out the switch on the Green/Brown wire(12volts). If the voltage is lower then the switch is faulty and if it's the same as on the Green wire then check at the rear lamps on the Green/Brown wire.If the voltage is low it shows a bad connection in the wiring between the switch and tail lamp plug.
Hope this helps.
cheers,
Stevie.