I really think you need to put the multi meter in the cupboard. Forget voltages. Just because the meter can read them doesn't mean it's information you need to know.
Test light is black and white. Power or no power. Is there power at fan? Yes? Then there is an issue with a component. No. There is an issue with wiring.
Bypass the reverse switch. Do you now have power to lights? Yes? It's a light fitting or switch issue. No? It's a wiring issue.