It's a funny old world, isn't it.
I suggest getting an electrician to extend the circuit from the existing socket by crimping new lengths of cable on, put a socket somewhere else and replace the old socket with a blanking plate, and the OP tells me that didn't help.
londonboy says "I'd get a local electrician to move the socket over, to the side of the book case, using mini trunking (which the bookcase will hide), which involves no chasing out, and probably just 2 hours labour (subject to existing earthing and bonding status)" and the OP thanks him for a useful post.
I asked if the bookcase goes all the way to the ceiling, to which the answer was "The bookcase does go all the way up to the ceiling", so I didn't pursue the option of raising it up (although my method would have used wooden battens and legs to match the bookcase).
OwainDIYer asks "Could you raise the bookcase on bricks so you can reach under it to get to the sockets?" and the OP thanks him for a useful post.
Perhaps you need to re-read your posts. They weren't particularly helpful as they implied I wasn't reading what others had written, in fact it was you who hadn't read what I had written. You also weren't the first to suggest putting a socket elsewhere.
I thanked the other posters because they were trying to help me in between all the nonsense posting from you and John (who thankfully has given up advising me to ruin my bookcase and removed his answers). Incidentally I did thank you for your original post but since it wasn't welcome I will revoke that.