Minis with remote gear change, including early Clubmans (not rod change, as on later Clubman) used the switch that fitted under the oil filter on the front face of the gearbox. Later Clubmans with rod change gearboxes had the reversing switch under the gear lever mechanism as winabbey's photo shows.
This is the switch used on the gearbox:
viewtopic.php?f=27&t=100890Under the switch, there is a spring AND a distance piece. The distance piece pushes against the 'actuating tit' when reverse gear is engaged.
The spring is the standard detent spring to put tension on selecting reverse gear; the distance piece turns the reverse light on.
gtogreen1969 has come up with more details.
The original switch was part number GAE131 but now that part number is used by a similar looking switch but the 'tit' and thread size is too big for the mini.
If your reversing light was working, you must have the distance piece 'somewhere'.