LONDON — On the evening of Dec. 21, Prime Minister Boris Johnson appeared from 10 Downing Street to tell anxious Britons they could “go ahead with their Christmas plans,” despite a surge in new coronavirus cases. At nearly the same moment, President Biden took to a White House podium to give Americans a similar greenlight.

It was a striking, if unintended, display of synchronicity from two leaders who began with very different approaches to the pandemic, to say nothing of politics. Their convergence in how to handle the Omicron variant says a lot about how countries are confronting the virus, more than two years after it first threatened the world.

For Mr. Johnson and Mr. Biden, analysts said, the politics and science of Covid have nudged them toward a policy of trying to live with the virus rather than putting their countries back on war footing. It is a highly risky strategy: Hospitals across Britain and parts of the United States are already close to overrun with patients. But for now, it is better than the alternative: Shutting down their economies again.

“A Conservative prime minister trying to deal in a responsible way with Covid is very different than a Democratic president trying to deal responsibly with Covid,” said Geoff Garin, a Democratic pollster in Washington. And yet, he said, their options are no longer all that different.

“From both a medical perspective and a political perspective,” Mr. Garin said, “there’s not as strong an imperative for people to hunker down in the way they were hunkering down a year ago.”

Some analysts say the two leaders had little choice. Both are dealing with lockdown-weary populations. Both have made headway in vaccinating their citizens, though Britain remains ahead of the United States. And both have seen their popularity erode as their early promises to vanquish the virus wilted.

Several of Mr. Biden’s former scientific advisers this week publicly urged him to overhaul his strategy to shift the focus from banishing the virus to a “new normal” of coexisting with it. That echoes Mr. Johnson’s words when he lifted restrictions last July. “We must ask ourselves,” he said, “‘When will we be able to return to normal?’”

Devi Sridhar, an American scientist who heads the global health program at the University of Edinburgh, said, “The scientific community has broad consensus now that we have to use the tools we have to stay open and avoid the lockdowns of 2020 and 2021. But it’s not easy at all, as we are seeing.”

The alignment of Mr. Johnson and Mr. Biden is significant because Britain has often served as a Covid test case for the United States — a few weeks ahead in seeing the effects of a new wave and a model, for good or ill, in how to respond to it.

It was the first country to approve a vaccine and the fastest major economy to roll it out. Its frightening projections, from Imperial College London, about how many people could die in an uncontrolled pandemic helped push a reluctant Mr. Johnson and an equally reluctant President Donald J. Trump to call for social distancing restrictions in their countries.

That Mr. Johnson and Mr. Trump initially resisted such measures was hardly a surprise, given their ideological kinship as populist politicians. When Mr. Johnson locked down Britain, several days after his European neighbors, he promised to “send the virus packing” in 12 weeks. Mr. Trump likewise vowed that Covid, “like a miracle,” would soon disappear. Both later suffered through bouts with the disease.

Mr. Biden, taking office, promised a different approach, one that paid greater heed to scientific advice and embraced difficult measures like “expanded masking, testing and social distancing.” Though Mr. Johnson never flouted scientific advice like Mr. Trump, he was sunnier than Mr. Biden, continuing to promise that the crisis would soon pass.

But both he and Mr. Biden have languished politically as new variants have made Covid far more stubborn than they had hoped. Last July 4, with new cases dropping and vaccination rates rising, Mr. Biden claimed the United States had gained “the upper hand” on the virus. Weeks later, the Delta variant was sweeping through the country.

In England, with nearly 70 percent of adults having had two doses of a vaccine, Mr. Johnson lifted virtually all social-distancing rules on July 19, a bold — some said reckless — move that the London tabloids nicknamed “Freedom Day.” After a midsummer lull in cases that appeared to vindicate Mr. Johnson’s gamble, the Omicron variant has now driven new cases in Britain to more than 150,000 a day.

Mr. Biden and Mr. Johnson have different powers in dealing with the pandemic. As prime minister, Mr. Johnson can order lockdowns in England, a step he has taken twice since his first lockdown in March 2020. In the United States, those restrictions are in the hands of governors, a few of whom, like the Florida Republican Ron DeSantis, have become vocal critics of Mr. Biden’s approach.

For Mr. Johnson, the major obstacle is not defiant regional leaders or the opposition but members of his own Conservative Party, who fiercely oppose further lockdowns and have rebelled against even modest moves in that direction.

The prime minister has kept open the possibility of further restrictions. But analysts say that given his eroding popularity, he no longer has the political capital to persuade his party to go along with an economically damaging lockdown, even if scientists recommended it.

Mr. Johnson is “essentially now a prisoner of his more hawkish cabinet colleagues and the 100 or so MPs who seem to be allergic to any kind of public health restrictions,” said Tim Bale, a professor of politics at Queen Mary, University of London. They “just feel that the state has grown too big in trying to combat Covid and that they really don’t want the government to grow any bigger,” Mr. Bale said.

Some British analysts draw a comparison between red-state governors like Mr. DeSantis and Conservative lawmakers from the “red wall,” former Labour strongholds in the Midlands and the north of England that Mr. Johnson’s Tories swept in the 2019 election with his promise to “Get Brexit done.”

These are not low-tax, small-government conservatives in the tradition of Ronald Reagan or Margaret Thatcher, but right-leaning populists who model themselves on Mr. Trump and the Mr. Johnson who championed the Brexit vote — voters the prime minister would need to win re-election.

Some critics argue that Mr. Biden and Mr. Johnson are both out of step with their countries. Britons have proven far more tolerant of lockdowns than the lawmakers in the prime minister’s party. In parts of the United States, by contrast, popular resistance to lockdowns is widespread and deeply entrenched.

“Biden suffers from seeming to do too much and Boris suffers from seeming to do too little,” said Frank Luntz, a Republican strategist who was a classmate of Mr. Johnson’s at Oxford University. “Biden would have done a better job if he had led Britain, and Boris would have done a better job if he led the U.S.”

Mr. Biden, unlike Mr. Johnson, does not face an internal party rebellion on his Covid policy. But the continued grip of the pandemic has sapped the president’s poll ratings, stoking fears of a Republican landslide in the midterm elections. The calls for change from members of Mr. Biden’s former scientific brain-trust, some said, reflected concerns that his Covid messaging was lagging reality.

Others pointed out that the president’s determination to keep schools and businesses open, despite the soaring number of cases, signaled that a change in thinking was underway in the White House — if a few months later than that in Downing Street.

“When Biden says we ought to be concerned but not panicked, he’s meeting Americans where they are,” Mr. Garin, the Democratic pollster, said. “He’s also meeting the science where it is.”

Stephen Castle contributed reporting.