Put the ignition on and check that one of the two pins at the switch plug is live, if it is then either the switch is duff, or there is a break between plug and rear lights, there is no relay for the brake lights.
Using a metre is ok but this will miss things ie it can easily read ok on a multi metre esp a digi one, but not have suffecient conductivity to pass current under load, brake lights are going to use roughly 3.5 amps, a multi metre passes a huge amount less.
If no live present with ignition on then check fuse 21 very carefully pull it out, due to age and the use of aluminum for the blades of the fuses these corrode, often just a clean up with a bit of emery will sort things, of course one of the terminals the fuse goes into will be live with ignition on.
tabetha