Check all your wires behind that panel with the ignition button on it. On my 07 Exige S, the wiring is weird. I had to fish out a wheel well bolt spline that got pushed into that area (I actually found it). All of the wires are tagged, but there are more male than female, or the other way around. I do not have back up lights in my car (I don't know why!!!), but everything else works.
I will email you the service manual, which includes a lot of schematics. You did not mention the make and year of your car OR I am too lazy to find it
If the schematics don't help or don't include your year and model, let me know and I will take some pictures of inside that panel and of my rear lights and harness. This would be for a 2007 Exige S. I can also take pics of my dad's 2005 Elise if this would be a closer match.
Let me know how things pan out. I feel your pain... Why doesn't my exige have backup lights?
