Okay... If the reverse light does not activate when moving the shifter to reverse means that the fifth/reverse lever down the side if the transmission (on the inside) is not moving the switch. You can remove and test that reverse switch just to be sure it works, but, if gears 5 & R aren't engaging there is more going on than the switch for sure.
I have seen this many times. If the 5-R lever is not working then the clip has probably come off of the end of the pivot stud in the inside of the case. This is the big T0RX-50 bolt you see on the side of the case. Most of the time this happens when someone works on the car and for some reason thinks they need to remove that TORX Bolt. It all goes downhill from there. Once the inside pieces attached to that pivot bolt get dislodged, the transmission has to come apart to get things back together correctly.
The story I often hear is that someone wanted to check the oil in the transmission and thought that would be the place to check it...
Perhaps that clip just popped off. That would be rare, but still possible. Without the clip, the lever won't stay in place and will no longer align with the pieces that it needs to make contact with in order to engage those two gears. The lever would also not remain close enough to the reverse light switch to operate it with the clip missing.
One might think that the 5-R Fork has broken in the tail housing, but in that case, the lever would still probably be in position and the reverse switch should still work the lights when the shifter is moved to reverse.
Stan