LONDON — Despite being headlined by a genuine star and staged at one of London’s premier theaters, a play about the foundation of a sprawling and troubled public service seemed unlikely to provoke night after night of standing ovations. But that’s what happened with “Nye,” an unlikely hit about the creator, and origin story, of Britain’s taxpayer-funded National Health Service. The play, written by Tim Price and directed by Rufus Norris, came at an inflection point for the NHS, as it’s known. Almost 80 years after it was founded, the medical service once touted as the envy of the world...