Why guess, when you could get some data?

In the UK, the per-capita spend on healthcare is $3480, of which $2919 is paid by the government (and so raised via taxes) and $561 is private (via health insurance or direct payment).

In the US, the per-capita spend is $8362, of which $4437 is paid by the government (via taxes) and $3925 is private (via health insurance).

This is only a cursory glance, but it does look a bit like nationalized healthcare results in lower private spending and lower government spending (the extra spend in the US presumably goes into the pockets of doctors, pharmaceutical companies and insurance companies).

Data sourced from the World Health Organization, via The Guardian[0]

[0] http://www.theguardian.com/news/datablog/2012/jun/30/healthc...