High night time electric consumption?

Retire

Isn't an immersion wired into a 13A plug and socket less than ideal?

I don't know about now but years ago it used to be common practice to have the immersion on a 15 amp plug and socket so that nothing else could be plugged into that socket.

Perhaps yours was like that and someone replaced the plug and socket with a 13 amp plug and socket thinking they were upgrading the plug and socket.

I remember when we moved into a new house and had to buy new plugs for everything because we had 5 amp and 15 amp plugs and our new house had 13 amp sockets. My dad wasn't happy about having to buy all these new fangled plugs.