All new bathroom supplies need RCD protection, also England and Wales the bathroom is a special location, so were it not for the virus you would need to either use a scheme member electrician or register the work with the local authority building control, I would think at the moment that would be difficult, but not sure if it would be considered emergency work, for emergency work you are allowed to register the work after the event.
So the under floor heating needs to be the special type with an earthed braid or covered with an earthed mat, it is special for bathrooms, again the other equipment must be suitable for bathroom use, there are what seems silly rules, like needing to use a tool or key for access, so shower pump under bath with screws holding on bath sides is OK, but if no screws in bath sides then same as if not under bath.
Even what forms a bathroom, never understood why shower panels are not 2.25 meters high, if they were then the room would not be the bathroom only inside the shower cubical would be classed as a bathroom.
So 850 watt is only just over 3.5 amp so load wise it is not really going to be a problem, and a RCD FCU with a 5 amp fuse would be good enough to comply with RCD requirements if the CU is not already protected, of course since a bathroom testing is important, and you will need to complete a minor works certificate, these can be down loaded from the IET web site, and if your able to complete the form, I would say the installation is safe enough, and I think after a set time you don't need to inform the LABC, and to be frank I moved house last year and could not find the certificates so applied for replacements, and council could not provide them, so much for traceable records. I did find them in the end.
You have said "Can I" and since I have no idea of your ability I can't say if you have the skill, all I can do is advise on laws and regulations, and officially the answer is yes, but it will cost more than getting a scheme member to do the job, last time I looked to inform the LABC cost £100 plus VAT, so it was not worth doing it DIY and staying legal, of course you can break the law, but your not I assume asking can I break the law, your asking can I within the law?
I personally don't care if you break the law, but I do care if you injure your self or others, so you need a low ohm meter and an insulation tester, I bought one of the latter when I misplaced my normal one, it cost I think £35, it actually tests using 500 volt, it is not a multi-meter.
I want to see you safe, so even if I think you are breaking the law will answer questions.
As to under floor heating, I fitted some in my late mothers wet room, it was a waste of money, it took 2 hours to warm up, and once all the insulation was fitted under the floor the floor was not what one would call cold without the under floor heating, 27°C is maximum floor temperature, and in a bathroom you want room at 22°C so just 5°C difference, that is not enough to heat the room. Lucky also a towel rail connected to central heating, but with the fan running drawing in air from the hall, it needed a bit more, I would turn off the fan.
It was over seen by the LABC and because it was for my mothers disability no charge, but just as well as after the first visit we never saw the building inspector again, the charge is little more than a tax.
Thanks, Eric, a very useful reply. So - RCD required. The shower pump is in a framed cupboard - access panel without handles. The heated floor is more to ensure that the wet shower floor dries out, rather than providing heating. I am aware they take quite some time to heat up - but they are effective in dispersing the condensation, and a fan in the room is essential to get the moisture out of the room.
In last 10 years I have lived in three houses, with one we had mould problems, not the other two, so I sat back and tried to work out why only the house in Mold had a problem with mould.
I realised it was having the shower in the bath, the shower glass panel has a gap top and bottom, so when having a shower the air circulates in at bottom out at top so whole room gets damp, but wet room has no panels so moisture did not circulate and shower room the glass panels seal at bottom so again moisture does not circulate.
To use a fan seems great, but with the Shotton house we found the air drawn from the hall is much cooler than rest of house, and so it ended up being switched off, turning it on when leaving room to dry room out great, but while having a shower if there is a free flow of air to replace that sucked out by fan, then that replacement air needs to come from some where warm.
In the house in mid Wales the daft builder did not replace the vent when fitting water proof wall boards in shower cubical, so we expected a problem, but it has proved there is non, there is only a gap one side of shower cubical at top and air does not circulate so rest of room stays dry.
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