Reverse lights work off your reverse switch on your gearbox (transmission). It is possible that the switch is faulty or sensitive to a specific voltage. LED's generally require an inline stabiliser to work. If it says plug and play then it should work.
Neutral safety switch won't do anything for your reverse lights. If you can get to the reverse switch and you're handy with a multimeter, you could try checking the voltage coming from the switch with ignition on, engine off and in reverse. If not showing any voltage i'd suspect the switch. If showing voltage. Go to the tail lights and see what voltage is showing at the reverse light plug. If nothing suspect the wiring between the switch and plug.
Are there fuses under the bonnet (hood) also check those too.