[David G4EBT]

It looks to be powered from one of those small Bakelite bell transformers which have several voltage settings, with the mains via an electric clock socket. If the bell works fine from the front door, the only thing that can stop it working from the back door is the voltage drop. Hence, it's my guess that the back door will be further away from the bell than the front door, and that the voltage drop from the transformer secondary caused by a long run of twin wire bell flex will lower the voltage to a point at which the bell solenoid is insufficiently energised. Worth checking if the bell is connected to correct tapping on the transformer - which I assume it will be, then cleaning all the connections to the bell push and the bell to reduce the resistance of the connections.

Hope that helps.