LONDON — Britain’s National Health Service, put in place by the country’s post-World War II Labour government, holds a unique place in the country’s psyche as both source of constant frustration, object of affection and — somehow — a central pillar of arguments both to leave and remain in the European Union. In a country riven over Brexit, at least most people can agree on the importance of the N.H.S.

What its future should be is less clear.

The N.H.S. suffered from staff shortages and tight budgets long before Brexit, but in the year following the referendum almost 10,000 nurses quit. In November, Simon Stevens, the chief executive of the service in England, said that after seven years of budget constraints “the N.H.S. can no longer do everything that is being asked of it.”

Amid a flu outbreak, colder weather and the continuing shortages of beds and staff this winter, some hospitals declared “black alerts,” meaning they were unable to deliver comprehensive care. Nonurgent surgeries were postponed and patients waited more than 12 hours in emergency wards before being tended to — undergraduate medical students were reportedly asked to volunteer to ease some of the strain.

In early January, Prime Minister Theresa May was asked if the service was in crisis, and responded that it was better prepared for winter than “ever before.”