With a multimeter on a DC voltage scale you should as you note get around 13.5 volts with the engine running.
It appears your tester did not test the battery properly to do that, they use a device with a resistor in it to take 100 amps or so from the battery and test the voltage when the battery is doing that. It would be below 12.5 volts after a short period.
The next action you need to take is to clean all connections and tighten them especialy at the dynamo if you have not got an alternator and at the voltage control box if you again do not have an alternator.
Checking for a tight fan belt, I assume has been done.
After that a read at this MG pdf will show what is needed next but you may want an auto electricion to do the work.
http://mgaguru.com/mgtech/books/pdf/Lucas_Generator_and_Control_Box_Tests.pdf
