The over heat is because the fuse (13 amp ) is NOT limiting the total current of both appliances to current that the assembly of three sockets can safely carry.with a 13A fuse, thus limiting your available power and causing the overheat.
Interesting. You need to go back to the consumer unit (fuseboard) and look at the circuit breaker (or rewirable fuse) supplying that socket.
Trouble is, that socket could be a spur from a ring final (which is fine), could be the end of a radial (which is fine) could be a spur from a spur (not fine), could be on a lighting circuit for all we know.
I just turned the mains switch off in the fuseboard to take the face plate off.
Anyway these are the two for the sockets in the house.
Does that help?
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