The US has never had a 'national' healthcare system comparable to the NHS, so it isn't that it was 'privatised' as such. The US took the opposite path to the UK in the mid-20th century; where the Attlee government decided that healthcare was primarily a state concern, the US instead entrenched private health insurance as the cornerstone of its healthcare system.

Part of that is a question of scale: the challenge of creating the NHS in Britain pales in comparison to the task of creating a universal, state-run healthcare system in a country the size of the US, especially when you take into account the ever-vexed question of states' rights and whether the federal government would actually have legal competency to do so.

But it's also a historical quirk: healthcare was excluded from the Depression-era Social Security Act, and although there were efforts by the FDR administration to reform healthcare, they were stifled by other priorities. For a quick overview of the chronological evolution of US healthcare policy, this PBS timeline is pretty useful.

During the Second World War, private health insurance (which originated in the 1920s with the creation of the Blue Cross system) became an increasingly common perk offered by employers to attract talent, to such an extent that it became a widespread standard benefit — and, as you'd expect, the private insurance industry flourished as a result.

Thereafter, the debate over healthcare reform has always tended to focus on who provides the health insurance — not on whether the state itself should operate the healthcare system. So the major federal programmes, Medicare and Medicaid, don't provide healthcare for the poor and elderly; they provide health insurance. It's worth noting that Britain's system, in which the majority of doctors are actual or quasi-state employees, is relatively unusual.

The fundamental premise of the Affordable Care Act (Obamacare) — the individual mandate which makes having health insurance compulsory — isn't new; there were campaigns for a similar law at various times during the 20th century. This lecture from a member of Physicians for a National Health Program (have a guess at their agenda...) gives a decent quick overview of those efforts.