I think the way forward is compulsory health insurance for the day to day stuff with the NHS being an emergency service provider and provider of Children's healthcare. This would need to be planned for and introduced over an extended period to allow people to adapt to what is a cultural change. A lot longer than the term of a government, possibly 10 or more years. There must be an affordable insurance scheme, that you can only opt out from if you have a better scheme in place. There also needs to be time for the private healthcare system to expand to take up the slack, and existing services to introduce the finance systems for a half and half model.service
There should be no access to the free part except in a genuine emergency for those visitors to the UK without insurance, where there is not a reciprocal treatment arrangement between their government and ours in place. It should always have been emergency treatment and send home, it just didn't get enforced. Staff who tried were branded inhumane and or racist.
As for the non contributing members of society it should be possible to fund a basic insurance with adjustments to the benefit system. Covering the cost of basic healthcare insurance for that section of society should be cheaper than running a totally free service.
Yes I know this will lead to increased social division, those who can afford it may get better care, well if you choose to spend your money that you have earned on better healthcare you probably do already or why shouldn't you. You almost certainly have better housing, a better car and your children may go to better schools. I am sure I will be attacked for this, but sometimes when something is broken you have to pay for a replacement.