British electrical appliances (240 volts) can be used without problems. If you want to use devices designed for 110 volts, make sure they can tolerate 50 hertz (most shavers can't, radios can) and get yourself a transformer. If it's just the hairdryer or the iron that you can't spare, go buy one in Corsica, that's cheaper.
You don't have electricity available in mountain refuges. They do have electricity (solar panels and accumulators), but the current is 24 volts DC and is not accessible. So forget about taking that electric heater of yours with you.
If you want to take a TV set with you, make sure it is capable of digesting the French Secam signals, not PAL, not NTSC, and not even Secam East! Radio stations use the same bands and modulation types as everywhere else in the world, that is FM 88-108 MHz, AM 520-1620 kHz. On the continent, there are lots of radio stations broadcasting on short wave and long wave as well.
You'll find all types of batteries on Corsica, but of course not in the villages. If your appliances use something very fancy, take replacement batteries with you.