I think America has historically had a different culture towards health care provision. The ideal of which I guess is that it funds innovation, and for the the very wealthy provides superlative healthcare. I think though even very poor countries like Cuba, have shown a more proactive and preventative health care system means, life expectancy can be greatly increased and health care costs lowered.
I think that's the road the UK should be looking for, spend money to save money - the byproduct being to provide the same or better level of health care with better outcomes, at lower cost. The NHS budget for 2015/16 in the UK was £116 billion -
http://www.nhs.uk/NHSEngland/thenhs/about/Pages/overview.aspx
Just at a crude guess based on population that would be approx 5 x for the US so roughly $600-$700 billion. That's end to end though, and my guess is a national health care plan, at least covering the basics, could be implemented in the US at a much, much, lower figure. I think the barrier is far more the insurance companies opposition than financial. Though I do think some of the costs being levied for drugs now - may need regulation, where it is clearly exploitative profiteering.
In terms of finance, it would pay for itself, in many ways. It provides a level of social stability, a healthier workforce for employers, and makes the country more attractive to invest and employ people in. I think that would be part of the route to take - are there any ways could it lower costs and regulation for employers? Anything that can make a new plan more attractive to those who might have the power and influence to actually get it done. I think America is the only one of the G7 countries, not to have such a system. To me it needs a bi-partisan fact based plan to really make it happen.
That's just an outsiders perspective, and in no way would i endorse copying anything else from Cuba - apart from their preventative care approach.