Likely the answer is no, I say that because you have asked the question so likely don't have the skill, but I will go through it anyway.
Step one is to find what feeds that cable, for all you know if may have been for a thermostat, so until you know what feeds it, can't really move forward.
If it does have a good feed then the next is it is a special location so you would need likely to inform the local authority building control and pay their fee, which is normally over £100 so it would be cheaper to use a scheme member electrician.
Of course you can break the rules, so you could physically fit one, but the question one assumes asked if allowed, you can drive down the motorway at 120 MPH we see reports in the paper that people have done that, so it is possible, however it is a costly exercise and dangerous.
My question has always been why do you want a shaver socket today? Years ago we would shave using a mains shaver in front of bathroom mirror, today most shavers are battery, some rechargeable, and the problem with rechargeable is many shaver points are not continually rated, so you should not charge anything from them, and if rechargeable there is no need anyway. Same with the tooth brush, if rechargeable it does not need recharging in bathroom.
So a shaver plugged in to shaver point by moving the shutters switches on the transformer, likely the shaver is in your hand, so unlikely to fall into bath or sink, but anything being recharged is left unattended, so it can fall into bath or sink and not be seen, because it comes from an isolation transformer the RCD will not trip, current is limited to 100 mA by the transformer but it will not auto switch off. The theory is both supply wires are within the same device so current will only run inside the device, and not from the device to anything else including earth. This assumes the cable is A1, any damage to cable then current can flow outside the device.
I know you can get shaver sockets with the tooth brush logo which means you can charge items from it, but why?
So to have room for the isolation transformer the shaver socket will be larger than the back box you have so it will need changing but is could be as simple as just connect the wires you have and it will work, whole idea of shaver socket is safety, so should you really DIY?
So in same way as you can drive at 120 MPH on public roads, you can DIY fitting of a shaver socket, and to be frank less likely to get caught, but why?
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