should have wire 110v  circuit with  an earth leakage protection device, tripping current 30mA  or less, otherwise power service provider will not be allowed

should have wire 110v  circuit

Not this side of the pond! 230V is normal in bathrooms in the UK, although you're right about 30mA RCD for additional protection for all circuits.

Is it possible/allowable to spur the LED Mirror and shaver socket from the power socket?

Usually yes, I'd expect the shaver socket to be OK directly on the socket circuit, but the light will probably need fusing down (typically a 3A fuse in a fuse connection unit) - do but check manufacturer's instructions for both, as some may differ. One spur feeding both isn't a conventional circuit arrangement, but provided the loads are small (as they seem to be in this case) it should be physically fine. The new FCU wants to be outside zones - preferably in the other room (alongside the socket?). Unless the shaver socket is more than 2.5m away from the shower tray (or 3.7m from the shower head if it's a wet room) it'll need to be of the isolating transformer type. Check too that the light manufacturers state it's suitable for bathroom use (and for whichever zone it ends up in).

If the new work is within zones (e.g. <600mm horizontally from the shower tray) it will be notifiable work under building regs (part P) (in England/Wales at least) - which can be surprisingly expensive. In any event, to comply with BS 7671 you'd need to be able to test and certify your work too.

   - Andy.

AJJewsbury said:

Not this side of the pond! 230V is normal in bathrooms in the UK, although you're right about 30mA RCD for additional protection for all circuits.

I believe bathroom/shower room, at least some circuits under NEC (USA) require 10 mA GFCI protection - certainly socket-outlets, although use of an isolating transformer is not needed where 10 mA GFCI is used?

AJJewsbury said:

Not this side of the pond! 230V is normal in bathrooms in the UK, although you're right about 30mA RCD for additional protection for all circuits.

I believe bathroom/shower room, at least some circuits under NEC (USA) require 10 mA GFCI protection - certainly socket-outlets, although use of an isolating transformer is not needed where 10 mA GFCI is used?