So upon turning the key you just hear a click, the engine does not turn over. Motorbill66 is right, what you are most likely hearing is the starter relay. It is located on the passenger
side engine compartment, just behind the bonnet prop attachment. For a 1977, the original relay would have been round with four wires attached. Most relays have been replaced with the square "Bosch ice cube" relay.
The wire colors are black, white/red, white/brown, and brown.
Black = ground.
white/red = ign switch
brown = positive, always hot.
white/brown = relay to solenoid.
I note you have already tested the white/brown at the solenoid. That seems to be working correctly.
The way the circuit works is as follows:
When you turn the switch, power is supplied via white/red wire to the relay. With power to the white/red at the relay, the relay closes, and the brown wire supplies the white/brown power to the solenoid. The solenoid engages the starter and also supplies the white/light green 12V to the coil.
To be sure, disconnect the black wire or white/red from the relay, turn the switch. Sound gone? Power to white/brown gone?
As far as the problem goes, make sure you have connected the white/brown and white/light green wire to the correct terminal on the solenoid. The white/light green goes to the ign terminal on the solenoid. Original Lucas solenoids for the Lucas 2M100 starter have different sized terminals which match the MG's wire terminals. But a great many MGs now have aftermarket solenoids with the same sized terminals, easily crossed in this situation.
If not the above, then as enfoperfect relates; the battery is low or bad connection, or my personal favorite a bad solenoid.