In Wales, the letters were sent out by the Chief Medical Officer...(?) with Public Health Wales, who compiled their list based on specific diagnoses in digital charts within NHS Wales. People who are extremely vulnerable, but whose specific diagnoses didn't contain the keywords that NHS Wales searched for so they got missed out were instructed to contact their GPs, who could then check the file manually to confirm the person belongs on the list, then the GP surgery could contact Public Health Wales to get the person added.
However, as far as I can tell, supermarkets in Wales are not using the list to let those people book slots. The system here isn't quite as streamlined as what England appears to have... even though it's the same supermarkets, and there is a list here.
Right now, I am on the list, so if I need to sort out supermarket slots or some community-based support, I can. But as long as my husband remains healthy, we are going to carry on as we have been. There are people who need the support more than we do.