OR switch for bathroom radio
The best thing you can do is to use relays.  Some things to note about your original idea that make it not workable:


There is no such thing as a mains voltage AC OR gate.
You mustn't* run a radio directly off a low current lighting circuit.


So you will need to use the lighting circuit to control a separate higher current (ring main) power source to the radio.  The simplest circuit would be with two AC relays, one in parallel with each light, which feed power to the radio:







simulate this circuit &ndash; Schematic created using CircuitLab



*in most sensible countries