Well - small world. I have the inverse setup - old C&W connections, and I'm with BT. Various sockets dotted around the house and chopped-off lengths of cables and DIY extensions. I can join the gaps though - just need to keep the same colour wires together, otherwise weird things happen
The master socket you can touch. There are two screws on the front - these can be undone. There are two screws inside - these cannot be undone. This would be bad. When you remove the first set of screws, the front unplugs and inside is a connector for the extensions. It sounds like you need to move the wires from the BT socket to the NTL socket.