The NHS is the greatest social invention in the history of mankind. It's a beautiful organisation.
Nobody sane argues that a basic education shouldn't be free and accessible to all. Why is health care any different?
Well for one thing, schools aren't operated as centralized as something like the NHS is. If someone is unhappy with schools in the City of Chicago, he's perfectly free to move to a suburb or wherever else he believes his kids will be properly educated. (Or pay for a private school or homeschool in some counties.)
A national single payer system such as the NHS eliminates competition entirely, makes its citizens dependent on government policy for their healthcare, and objectively produces worse results than a lot of other healthcare systems that are just as universal.
There's no contradiction in supporting universal access to healthcare and opposing a massive government institution like the NHS.
1. I realise a national health service is not the only method of universal health care.
2. Localising health care would lead to the same issues as localised public education. Better services for those who can afford to live is affluent areas, inferior underfunded services for people in underprivileged areas. Not everybody is free to move to where they wish.
2. Having a government run national health service gives the citizens the ultimate power in how it's run in that they directly elect the people who run it.
3. British residents are free to purchase private health insurance and get treated in privately run hospitals if they wish.