The NHS does have serious problems, but it's a LOT worse now than it was a few decades ago, and the two significant things that have changed are a) funding (per capita and accounting for inflation) is far lower and b) a lot of privatisation has happened.
Politicians - mostly the tories but Blair's government take a lot of blame too - have defined "not privatising the NHS" as only meaning keeping it free at point of use. But in the background, NHS trusts have been forced to sell land and buildings that they used to own and rent it back from the new private owners, and many areas have seen both staff and equipment privatised, from agency staff (where instead of hiring cleaners or nurses or whatever, they instead hire agency staff, where the hourly cost to the NHS is double or more what an employee would cost with most of that increase going to the agency companies not to the workers) to private hospitals (where instead of investing in a new operating theatre, or whatever, they pay to have NHS patients operated on in private hospitals), etc.
The NHS is far from perfect, but the lesson we learn from those imperfections shouldn't be that nationalised healthcare is bad, but that underfunding it and then using that underfunding to justify privatising lots of stuff in the background is not a good way to run a nationalised health service.
The root problem is that many politicians would like to see the NHS fall to pieces so that an American healthcare system can create lots of opportunities for companies to make money, but because the NHS is hugely popular it would be political suicide to make that an official policy, so instead they've taken this approach which not only creates these short term opportunities for companies to come in and profit as mentioned above, but also gradually erodes the it's popularity with the long term goal being that eventually it won't be political suicide to say "Look the NHS is a failed experiment, we need to replace it with American style private companies and healthcare insurance".