Comments from the front-runner for prime minister, Liz Truss, suggest bumpier relations with Europe and the U.S. if she wins, with Northern Ireland getting in the way of the “special relationship.”LONDON — Three weeks after Liz Truss became Britain’s top diplomat in 2021, she told a Conservative Party conference that her country need not compete for the affection of the United States. Britons, she said, should not worry “like some teenage girl at a party if we’re not considered to be good enough.”Her line drew laughs, but little more than that, at a meeting dominated by the flamboyant figure of Prime Minister Boris Johnson. Now, though, Mr. Johnson is on his way out and Ms. Truss is the front-runner in the contest to replace him, making such provocative comments a potential clue to future policy.Should Ms. Truss emerge victorious in a party vote that will be announced on Monday, she will have a chance to flesh out the vision of a Global Britain that Mr. Johnson unveiled after the country left the European Union two years ago. Based on her record as foreign secretary, diplomats and analysts in London and Washington said, relations could get bumpier with the United States and, even more so, with Europe.Tensions between London and Brussels have already flared over legislation introduced by Ms. Truss that would upend the post-Brexit trade arrangements in Northern Ireland. She has vowed to push the new law through Parliament, stoking fears that it could trigger a trade war across the English Channel.The Biden administration is keeping close watch, anxious that the dispute could threaten a quarter-century of peace in Northern Ireland secured by the Good Friday Agreement. President Biden has asked aides to pass along his concern about the negotiations between Britain and the European Union over the trade rules.Rishi Sunak met supporters last month in Birmingham, England. During the campaign, he has promoted his credentials as a Brexiteer.Rui Vieira/Associated Press“We’re going to trundle along in a pretty bad place” in part because “she’s going to keep playing to the peanut gallery of those who are deeply committed to Brexit,” said Leslie Vinjamuri, the director of the U.S. and Americas program at Chatham House, the British research institution.“There is a swath of Britain that doesn’t like being dependent on the United States or the European Union,” Ms. Vinjamuri said. “She is completely aligned with a vision of Britain being global, strong, sovereign and, most of all, independent.”That Brexit-inflected message has helped Ms. Truss pile up a commanding lead in the polls over her opponent, Rishi Sunak, even if he performed well in the final debates of the campaign. But some of the pressures will mount regardless of which candidate is victorious. Mr. Sunak, too, has pledged to push through the Northern Ireland bill, and he promotes his credentials as a Brexiteer. (Ms. Truss opposed Brexit before becoming a fervent proponent of it after the 2016 referendum.)Britain’s role in the world is shaped by forces larger than the next occupant of 10 Downing Street. Having cast itself off from the European Union, Britain can act as more of a free agent, seeking its own relations with great powers like China. But it has lost its role as a bridge between the United States and Europe, becoming a less influential player on global issues like Russia’s war in Ukraine.In that crisis and others, including Iran’s nuclear program, Britain is likely to keep aligning itself with the United States. Mr. Johnson has acted as a kind of wingman to Mr. Biden on Ukraine, encouraging him to impose harsher sanctions on Russia and ship heavier weapons to the Ukrainian army.Loyalists marching last year in Portadown, Northern Ireland, to protest trade rules that require border checks on goods flowing from mainland Britain to the North, as part of a Brexit deal with the European Union.Paulo Nunes dos Santos for The New York TimesMs. Truss would most likely double down on Mr. Johnson’s backing of Ukraine’s president, Volodymyr Zelensky. She has presented herself as a hawk on Russia, using language that at times goes further than that of American officials. But her most memorable diplomatic encounter, with the Russian foreign minister, Sergey V. Lavrov, in February, was marred when Russian officials claimed to reporters that she was ignorant of Russian geography in a private exchange with Mr. Lavrov.While Ms. Truss lived with her family in Canada for a year as a child, she is not a globe-trotting figure like Mr. Sunak, who has an M.B.A. from Stanford, owns a home in Santa Monica, Calif., and until recently held a U.S. green card. Mr. Johnson was born in New York City and renounced his American citizenship only in 2016.For all of his Brexit bluster, which appealed to former President Donald J. Trump but grated on Mr. Biden, Mr. Johnson regularly expressed his affinity for the United States. In his last appearance as prime minister in the House of Commons in July, Mr. Johnson offered his successor some parting advice, borrowed from his hero, Winston Churchill: “Stay close to the Americans.”Ms. Truss, by contrast, shows little reverence for the “special relationship” between Britain and the United States. “It’s special, but not exclusive,” she said at the party conference last year, noting that Britain had other important allies like Australia, India and European countries, notably the Baltic States.“Words matter,” Ms. Vinjamuri said, “and they matter especially when the U.S. is in a period of elections and political upheaval.”Prime Minister Boris Johnson of Britain with President Biden in June during a NATO meeting in Spain. In the final moments of his last appearance as prime minister in the House of Commons in July, Mr. Johnson offered his successor some advice: “Stay close to the Americans.”Jonathan Ernst/ReutersMs. Truss’s most ambitious outreach to Washington came during the Trump years and ended in frustration. As trade secretary, she led negotiations for a trans-Atlantic trade agreement with Mr. Trump’s trade representative, Robert E. Lighthizer. He recalled her as an energetic, well-briefed free trader.The talks, however, petered out with Mr. Trump’s defeat in 2020, and Mr. Biden has shown little interest in reviving them. That means Ms. Truss will have to find other common ground with him, beyond Ukraine.“Brits expect their prime minister to have a good personal relationship with the American president,” said Kim Darroch, a former British ambassador to Washington. “If they’re not getting on, that will get picked up and commented on, probably critically.”Perhaps stung by her previous statements about foreign leaders, Ms. Truss has steered clear of American politics. Asked by a journalist last week if she viewed Mr. Trump as a friend or foe, she said, “I’m not going to comment on future potential presidential runners,” adding, “We have to work with whoever is in the White House.”She was less diplomatic when it came to continental Europe, particularly France. “The jury is out,” she said, when she was asked the friend-or-foe question about President Emmanuel Macron of France. That drew a backhanded reply from Mr. Macron, who said Britain was a friend, regardless of its leader.Peter Westmacott, another former British envoy to Washington, likened Ms. Truss’s remarks to those of a candidate in an American primary — in this case, aimed at the 160,000 or so members of the Conservative Party who are voting for a new leader. If she wins, he predicted, she will pivot back to the center.Still, he said her campaign messaging had done damage that went beyond France. She floated the idea of Britain sending asylum seekers to Turkey in addition to Rwanda, a proposal swiftly shot down by the Turkish government.President Emmanuel Macron of France last month in Paris. He said Britain was a friend of France, regardless of its leader.Yoan Valat/EPA, via Shutterstock“I hope she will also conclude before too long that the U.K. has every interest in finding allies in Europe to help limit the damage caused by Brexit, energy prices and Russia’s invasion of Ukraine,” Mr. Westmacott said.A trade war with the European Union is far from inevitable. European officials may choose to hold off on major retaliation until the Northern Ireland legislation gets through Parliament. That process could drag on for months, given the fierce resistance the bill is likely to face in the House of Lords, where many members view it as a breach of international law and a power grab by cabinet ministers.The dilemma for Ms. Truss, if she wins, is that her political ascent has been powered by her cultivation of the party’s Brexiteer wing. That will make it hard for her to give ground in the dispute with Brussels. And Britain’s relations with the European Union are increasingly inseparable from its relations with the United States.“The U.K.-E.U. relationship looks more destructive in the short term,” said Mujtaba Rahman, an analyst at the political risk consultancy, Eurasia Group. “It’s going to subtract from the level of credibility she’ll enjoy in Washington.”“All roads run through Europe,” he said. More