Yes you could get power using SWA but not really that practical. As to working in a bathroom most countries and principalities with in the UK have laws which if obeyed means that DIY will cost more than getting an electrician in to do the work.
The problem is the councils minimum charge is often set for the first £1000 worth of work. So fee to nearly rewire whole house is the same as in Wales for fitting an extra socket in the kitchen. As said it does vary.
So in the main unless braking the law DIY in a bathroom is just not worth it.
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