In November, the chancellor of the Exchequer, Philip Hammond, announced an additional 350 million pounds, or $475 million, for the National Health Service in England, describing it as an “exceptional measure” to ease pressures on services during the winter. He said an additional £1.6 billion would be provided for 2018-19.

But that fell far short of the £4 billion in additional funding that Simon Stevens, the chief executive of the service in England, requested last year, warning that services would come under unprecedented pressure during the winter.

“The N.H.S. waiting list will grow to five million people by 2021,” Mr. Stevens said in an impassioned speech to health care leaders in November. “That is one million more people, equivalent to one in 10 of us, the highest number ever.”

What’s more, he said, “after seven years of understandable but unprecedented constraint on the current budget, the N.H.S. can no longer do everything that is being asked of it.”

Over the past week, hospitals have increasingly declared “black alerts,” an admission that they are unable to cope with demand, the health service confirmed, without releasing numbers. Most hospitals have been unable to meet emergency-ward targets of seeing patients within four hours because of a shortage of beds and staff.