My sister had an email from her electricity network provider (different to her electricity provider) asking if she needed to be on their priority services register. She told me about it, so I looked up my electricity and gas network providers (ie the people who put the pipes in the ground to bring the power to you) here:
energynetworks.org/customer...
Then I phoned each of them immediately and within minutes I'm registered as needing priority attention in the case of power cuts, outages, breakdowns etc. I didn't need to send any proof - or even tell them what my condition is - but it's wise to say 'yes' they can pass on your details to emergency services, British Red Cross etc locally, as it would be those local people who come to help you.
Of course, this doesn't help us pay for your heating ... being deemed eligible for winter fuel payments is still a battle to be had!
Please post the on any other networks for conditions that might be relevant. Thanks