This sounds like another job made for a multimeter (volt/ohm meter).
The first place I would start would be the neutral switch.
By disconnecting its wire and then using the meter set to read ohm's resistance, touching the bare end of the wire and the engine case with the test probes should show a reading.
It might take shifting the transmission into gear and back to neutral a few times to make sure the switch had been actuated but it should show continuity to ground.
If it did, while I was sitting there with the wires unplugged I would test the wire to the light. If the meter read zero, either the wiring has become disconnected or the bulb has burned out.
After checking the bulb I would know which problem to look for. A loose connection.