Reverse lights

Turn the ignition switch on, unplug the neutral safety switch, then use a stretched out paper clip or cotter pin to connect the two outer terminals in the switch connector. If you see a tiny spark, the lights should be on. That means the new switch isn't working. (Unlikely).

If the lights still don't turn on, check fuses next. Also check for 12 volts on one of those two outer wires in the connector for the neutral safety switch. If you have 12 volts on one of them, check for 12 volts on the bulb socket in the back. I can't remember when they switched from the round brass base bulbs to the flat plastic ones. If you have the round ones yet, check if they use a pushed-in ground terminal in the side of the socket that is corroded.

(I'll be back in a few hours).

Caradiodoc

Was the switch the problem?

There is no switch in the column. Current flows from the battery through the ignition switch, the neutral safety switch, to the light sockets, to ground. Somewhere in there after the ignition switch is a fuse. Do you have 12 volts on one of the two outer terminals in the neutral safety switch connector?

Caradiodoc