cars fitted with a reverse light from the manufacturer must have a working light come mot time.
as for the problem, if the fuse, switch and bulbs are ok, theres a problem with the bits inbetween. could be a loose/borken wire?
i dont see how a gearbox problem would cause the reverse light to stop working. you need to find out if the reverse switch is actually working first, because it could be fitted incorrectly, not working etc etc...