The NHS is actually organised as a large number of locally managed nodes. Doctor's practices (GPs) are self-governing to some extent, and hospitals are managed by trusts. There has been a lot of reorganisation recently (the Tories always reorganise the NHS, it is a thing they do).

My knowledge of the US is limited but I imagine that a US version of the NHS would be organised by state and then by district hospitals. Plenty of moving parts at the back end but at least the patient just sees the single payer front end.