I need an additional double wall socket in my living room. Can I drill through the lounge wall to the adjacent bedroom and connect back to back with the existing bedroom socket.

I see no reason not to do it like that.

I have several outlets that are back-to-back.

Just make sure to remove the socket on the other side and insulate the wires before you drill through

And if you have labels to indicate rooms served on power panel you might want to make a note of it (and on new outlet so you don't assume room breaker kills it).

I have done a couple of back-to-back sockets in my house. And since my walls are concrete I also had to drill/chisel out a hole for the other box---that was definitely the hard part...inserting/wiring-up the box to the freshly dugout concrete hole was the easy part. Guess I could have put a surface mounted box on and just drilled one hole to the other box to run the connecting wires, but I wanted the box flush mounted.

Also check the fuse rating and is it suitable for additional load. Depending on what you're going to use it for!! Also I would be inclined to make sure the proper socket wire has been used before you increase the potential load through existing wiring !