Without being able to see the wiring it's difficult to know what might be wrong. So when you switch it to the neck pickup only there is no output at all?

I'm sure you have all the volume and tone wound all the way up for testing, but worth double checking it's nothing silly like that.

From the description either the connection from the pickup to the volume pot or from the pot to the switch is faulty OR the pickup is not earthed and so not completing a circuit.

Is the wiring in the guitar? It's not unheard of for solder joints to break during that installation process, installing electronics in a hollow body is a bit of a mission, but you might need to pull it all out and post some nice clear pics for us to troubleshoot.