I’ll have to check this on a schematic but I know for a fact that my 78B’s brake lights only come on with the ignition on and won’t work at all with it off. I guess I could be wrong but I thought all MGBs where wired basicly the same way.
Back to the problem.
First things first – If you don’t already have one get a manual with a schematic or down load one from the internet for your particular vehicle. Then get yourself a basic digital multi-meter (DMM). I think Radio Shack has them for under $20. Good cheap investment anyway to have in your tool box - cause if you own an old car, you will evetually need it again.
Next check the fuses. If glass you should be able to notice if they are blown or not but if you ain’t sure use the meter set to ohms (resistance) and measure for zero ohms. Anything lower than 1ohm should be just fine. Assuming that all the fuses look good the real fun begins.
Using meter set it to dc volts, use the red (POS+) measure if any of the wires (lines) to the switch is hot (+12V) with the black (NEG-) test lead to chassis or ground. Depending if your car is a negative or positive ground you should read either +/-12 respectively. If you have twelve volts – you can use the meter’s Ohms (resistance) settings to check the brake light switch. If no voltage is detected trace back to the source.
With power off and the leads (or wires) disconnected - Measure across the switch while you engage or toggle it on & off. If you see zero (or close to zero) ohms across the switch when (on) closed then the switch is fine.
So if you have voltage getting to the switch and out the other side when it is on then you’ll need to trace back to the brake lights.
That should be enough to get you started. Try that and get back to us?
Good Luck!