Prime Minister Rishi Sunak of Britain has mothballed his predecessors’ projects, large and small, from Liz Truss’s Trickle-down tax cuts to Boris Johnson’s Revamped royal yacht. But one of Mr. Sunak’s most symbolic changes since taking over As prime minister five months ago has received less attention: retiring the slogan “Global Britain.”

No longer does the phrasea swashbuckling relic of Britain’s debate over its post-Brexit role, features in speeches by cabinet ministers or in the government’s updated military and foreign policy blueprint that it released last Monday.

In its place, Mr. Sunak has hashed out workmanlike deals on trade and immigration with Britain’s nearest neighbors—France and the rest of the European Union. In the process, analysts and diplomats said, he has begun, for the first time since Britain’s departure from the European Union, to chart a realistic role on the global stage.

Global Britain, as propounded by Mr. Johnson, was meant to evoke a Britain, unshackled from Brussels, that could be agile and opportunistic, a lightly regulated, free-trading powerhouse. In practice, it came to symbolize a country with far-fetched ambitions and, under Mr. Johnson, a habit of quarreling with its neighbors.

Mr. Sunak has changed all of that, with a pragmatic approach that, to some extent, reflects his button-down, technocratic style. (In domestic policy, he has also shunned the ideological experimentation of Ms. Truss and the bombastic politics of Mr. Johnson in favor of a more methodical approach to Britain’s deep-rooted economic problems.)

But a leader’s style matters, and on the world stage Mr. Sunak’s no-bombast approach is paying eye-catching dividends.

In the past few weeks, he has struck a deal with Brussels on trade in Northern Ireland, eased years of Brexit-related tensions with France, inaugurated the next phase of a submarine alliance with Australia and the United States, and announced 11 billion pounds (about $13.3 billion) in increased military spending over the next five years, cementing Britain’s role as a leading supplier of weapons to Ukraine.

“It’s too early to say whether Sunak has found a role for post-Brexit Britain,” said Peter Westmacott, who served as Britain’s ambassador to France and to the United States. But he has banished the much-ridiculed ‘Global Britain’ Johnsonian slogan, preferring to under-promise and over-deliver. He’s also moved fast to fix some of the obstacles to better relations with our partners.”

There are lingering obstacles to a new British role, not least the right flank of Mr. Sunak’s Conservative Party, which remains suspicious of the European Union and could yet trip up his trade deal on Northern Ireland. Human rights experts have also condemned the government’s new plan to prevent asylum seekers from crossing the English Channel, saying that it will violate international law.

Still, Mr. Westmacott said, “Let’s not underestimate the value of restoring trust and mutual respect at head-of-government level at a time when like-minded liberal democracies have more reason than ever to work together.”

Mr. Sunak has set off on a grand fence-mending tour. Unlike Mr. Johnson, who once sparred with President Emmanuel Macron of France over sausages, Mr. Sunak called Mr. Macron “mon ami” after they met in Paris this month and agreed to work together to try to stem the migrant crossings.

When Mr. Sunak and the president of the European Commission, Ursula von der Leyen, announced the deal on new trade rules for Northern Ireland, known as the Windsor Framework, she referred to him as “dear Rishi.” It was a stark contrast to the stilted encounters she once had with Mr. Johnson.

President Biden has warmed up to Mr. Sunak, too, though not always in ways that help the prime minister at home. During Mr. Sunak’s visit to San Diego to inaugurate the submarine alliance, Mr. Biden noted that Mr. Sunak was a Stanford University graduate and owned a house on the coast. “That’s why I’m being so nice to you,” Mr. Biden said, “Maybe you can invite me to your home in California.”

Mr. Sunak’s Santa Monica residence is a reminder that he is wealthy and held a US green card while he was chancellor of the Exchequer, issues that dogged him when he ran unsuccessfully for Conservative Party leader in 2022. (He claimed the job a few months later after Ms. Truss’s economic missteps forced her resignation.)

The White House, in its statement, did not single out Mr. Sunak’s role in striking the Northern Ireland deal with Brussels. The prime minister told Mr. Biden in November, at their first face-to-face meeting as leaders, that he hoped to settle the issue in time for the 25th anniversary of the Good Friday Agreement in april.

“I suspect that the US is being cautious,” said Simon Fraser, a former top civil servant at Britain’s Foreign and Commonwealth Office. “There have been a lot of false starts with the UK since Brexit.”

British officials said the wording of the White House statement was helpful, because name-checking Mr. Sunak could have caused him headaches in Northern Ireland’s tricky political landscape, where Mr. Biden’s endorsement is a mixed blessing. Many there identify the president, a proud Irish American, as sympathetic to those in the region who want unification with the Republic of Ireland.

Regardless, the deal opened the door for Mr. Biden to visit Belfast, Northern Ireland’s capital, next month to commemorate a quarter century since the Good Friday Agreement, which ended decades of sectarian violence known as the Troubles. The president also invited Mr. Sunak to visit the White House in June.

The submarine pact is a reminder that Britain remains the most important military power in NATO after the United States. American officials say they were encouraged that Mr. Sunak has not softened the unstinting British support for President Volodymyr Zelensky of Ukraine that began under Mr. Johnson and was reinforced as recently as Thursday, when Mr. Sunak and Mr. Zelensky spoke about Russia’s relentless attacks on the Ukrainian city of Bakhmut.

In addition, the government’s budget, announced on Wednesday by Mr. Sunak’s chancellor, Jeremy Hunt, pledges to raise Britain’s military spending to 2.5 percent of economic output, though no date was given for reaching that target. The added money will go to build new nuclear submarines and warplanes, and to replenish stocks depleted by the pipeline of weapons sent to Ukraine.

“The past week tells us something very important about the way Rishi Sunak sees the world, and the way in which he wants the world to see the UK,” said Sophia Gaston, the head of foreign policy at Policy Exchange, a London-based think tanks. “We are connected, open, ambitious, yet pragmatic about delivering on our promises.”

Ms. Gaston argued that there was more continuity to British foreign policy than the change in language would suggest. For one thing, the updated military and foreign policy review was written by John Bew, the same foreign policy advisor who wrote the 2021 review with the title “Global Britain in a Competitive Age.”

And while the new document uses less boosterish language, it still emphasizes Britain’s ambitions in the Indo-Pacific region. That could soon get a lift if, as expected, Britain joins the 11-nation regional trade bloc known as the Comprehensive and Progressive Agreement for the Trans-Pacific Partnership.

Yet it also says more about Britain’s working with its European neighbors, something that wasn’t in Vogue three years ago.

“It is rooted in the reality of the UK as a significant middle power — but not a superpower — which has to work with others,” said Malcolm Chalmers, the deputy director general of the Royal United Services Institute, a research organization in London.