Moving a double socket from its current position to a new position

It is fine to extend one double socket from another double socket as long as the cables are run within a cable zone horizontal or vertical from the existing) It is also acceptable to blank the existing socket and extend using Wago connectors to the correct rating of the circuit.

If you are looking to extend the socket to a new position must come across horizontally or up and down within prescribed zones, for the installation to be within regs the connections where the cable is extended must be maintainable. I suggest extending from current position within the wall (chasing will most likely be needed) and then covering the connections made in wagos with a Double Socket Blank plate for future access.

I would recommend getting a qualified electrician to carry out these works

The first thing you should get completed is a test at the socket you wish to alter, before doing anything to it you should prove it is electrically sound and safe. You will need the equipment and knowledge to do these tests. (ring continutiy, r1+ r2 figure of 8, zs, insulation resistance and RCD trip performance) . If you cannot do these then get an electrician that can. Moving the socket should be fine so long as the new cable runs in either the horizontal or vertical safe zones. You could use the old backbox as a junction box using Wago connectors with a blanking plate on the front. Wagos are certified suitable for the purpose and probably used by most competent electricians in this instance. You then need to retest as per the first paragraph at your new socket. I would get your electrician to provide you with a minor works certificate.