The UK NHI doesn't work well because the neoliberal parties in successive governments (both the Tories and New Labour) have been defunding it so that they can - like Thatcher did with the railways - once its quality has fallen due to lack of funds claim that it's bad because of Public management whilst it would be much better if it was Private because the Private Sector is much more competent, and privatise it.

Just like the US has fatcats that are perfectly happy to mass murder people for personal profit, so does the UK (and the British Political System is almost as bad as the American, so it's definitelly sold to the highest bidder) and plenty of those jhave wet dreams of the country having 13% of its GDP flowing through a Private Healthcare sector like the US were they can make billions of pounds doing exactly the same as the fatcats do in US Healthcare.

Source: I lived in Britain for over a decade.

By the way, you "read that the UK NHI doesn't work very well" is exactly because the UK media is overwhelmingly owned by tax avoiding billionaires who are part of the above mentioned fatcats who see themselves as profiting massivelly from Britain having a Healthcare System like the US. It's not by chance that the level of trust of Britons in their Press is one of the lowest in Europe.

The exact same kind of tactics were deployed by Tatcher back when she wanted to privatise the Railways with the result that satisfaction with the Railway system in the UK is now even lower than when there was a public operator even after Thatcher defunded it to claim "Public is Bad, Private is Good" to amass enough public support to privatise it.