I don't think her real issue is the price of healthier foods, she obviously has an unhealthy relationship towards food & I certainly can relate to that. I think perhaps the NHS could help her, but I'm thinking more down the therapy route.. Maybe a bit of CBT to help her to control her urges to binge? I don't think it's fair for her to be asking the NHS to give her money or food stamps etc so she can lose weight.. But there are other ways that they can help her, as they help everyone else - Such as prescription SW or WW groups, exercise on prescription (I believe you still have to pay some, but it's a lower price), therapy etc. These things would help her to not only lose weight but also to maintain a healthier attitude towards food as well.
I do find SW & WW expensive, but I think once you know where to look for food at the best prices, then it can be a lot cheaper! We ended up shopping in different supermarkets for different things, which saved us money!
I hope she does get the support she needs, rather than what she *wants* & is able to live a happier, healthier life. Atm, she is clearly a very unhappy person