You can use insulated crimps to join new cables to the existing ones.
You will need to use a decent crimp tool though, one with a ratchet mechanism is ideal. The cheap ones just can't make a good enough joint.
Before you do this you should check that any existing sockets in the room that will have the new one, are on the same MCB or fuse. If not then you will need to label the new socket to make that clear.
You should also make sure that the all of the cables end up in the "safe zones" for cables. If any of the wiring for your new socket goes outside that zone, then you will need to protect it in the usual way.
Details of the safe zones can be found at:
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you are happy to replace the existing socket with a blanking plate, then you could use suitably sized terminal blocks to join the cables rather than crimps.
You may find it easier to remove the existing socket completely and then install a new one in the other room.
If the existing socket is not a spur why not leave that where it is and have the new socket as a spur. If the new socket is on the same vertical line as the old you can drill through the wall behind the original socket and chase a groove down the other side of the wall for the cable without the need to provide protection as it will be vertically above the new socket.
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