Hi I add a closure note as the electrician came today.
It appears that it was the plug that was faulty, in particular the cable going to that plug: basically just underneath the plug/socket, the cable has been "eaten" by mice. The electrician suspected that it will even have a dead mouse behind the wood plank.
We were able to see some black "as burned" stuff on the wire/plastic of the wire. They were located a bit back on the cable going to the plug, he had to pull it firmly.
He could also hear some electrical noise when the fuse was on.
He put some "blank" closing socket so the plug will be inactive.
He also confirmed as the pro on this forum said, that it's illegal and dangerous to have a socket/plug on the shower.
At first he also talked about possible about a junction box, but he rejected that possibility as soon as he saw the "eaten charred cable".
Well as a closure note I guess it was useful to call the electrician, it did not seems to cost that much (he said something about £40/hour plus vat, invoiced to the landlord), and it's less dangerous than doing the fix on our own.
It seems to be all working well so far, 2 hours after he is gone!
Thanks all for your help it was great!