May be worth checking the wiring to see if it is damaged. So you know if the switch makes the circuit or breaks it, remove the plug and see if the reverse lights come on then try shorting the pins in the plug to see if they then come on. Do you have the correct counter balance weight fitted?
The selector was set up correctly?
Audi Workshop Service and Repair Manuals > A2 > Power transmission | 5-speed manual gearbox 02J | Gearbox mechanics,operation, construction,diff. | Exploded view - dismantling and assembling selector mechanism in gearbox
A2 > Audi Workshop Service and Repair Manuals > Power transmission | 5-speed manual gearbox 02J | Gearbox mechanics,operation, construction,diff. | Exploded view - dismantling and assembling selector mechanism in gearboxworkshop-manuals.com