As the Scottish nationalists stumble, a by-election near Glasgow this fall will test support for the opposition Labour party ahead of Britain’s coming general election.For Britain’s opposition Labour Party, the road to 10 Downing Street is likely to run through Scotland. And the first steps on that road lie in a cluster of commuter towns southeast of Glasgow, where Labour is trying to win over swing voters like Cara Scott, in a closely watched parliamentary vote that will test the party’s appeal ahead of a coming general election.Ms. Scott, 18, a geography student who studies in Edinburgh, enthusiastically supported the Scottish National Party in past ballots. But she is disillusioned by her latest S.N.P. representative, Margaret Ferrier, who was forced out of her seat on Aug. 1 after violating lockdown rules during the coronavirus pandemic.She also thinks the Labour Party has better proposals to cope with a grinding cost-of-living crisis that has left people fed up and exhausted. Ms. Scott signed a petition to recall Ms. Ferrier, which triggered this by-election, and now said she was “leaning slightly toward Labour, based on how proactive they’ve been.”“Their campaign has been brilliant,” said Ms. Scott, as she browsed in a slightly tattered shopping mall off the town’s high street. “Right from the get-go, they’ve been really trying to sway people’s voting opinions.”Cara Scott, 18, thinks the Labour Party has better proposals to cope with a grinding cost-of-living crisis that has left people fed up and exhausted. Emily Macinnes for The New York TimesIf the Labour Party can snatch back the seat, which it lost to the S.N.P. in 2019, it will be viewed as a harbinger of broader Labour gains across Scotland in the next general election, which the Conservative prime minister, Rishi Sunak, must call by January 2025.A Labour revival in Scotland could give the party the margin it needs to amass a majority in Parliament, even if — as most oddsmakers predict — its current double-digit lead in the polls over the Conservative Party narrows. A date for the election to fill the Rutherglen and Hamilton West seat has not yet been set, but it’s expected to take place in early October.“This will become the center of the political world in the U.K. for the next few weeks,” said Ian Murray, who holds the sole Labour seat from Scotland and serves as the party’s shadow secretary for the country.“If Labour wins the election in Rutherglen, you can say Keir Starmer is a prime minister-in-waiting,” he said, referring to the party’s leader, who campaigned in the district earlier this month. “It feels like the wind is at our back,” he added, “but if there’s any party that can fall over in the wind, it’s the Labour Party.”Labour has been reborn in Scotland by the same public distemper that is lifting it above the Tories south of the border (a Tory lawmaker, Nadine Dorries, quit last week in England with a venomous attack on Mr. Sunak, whom she described as leading a “zombie Parliament”). But this is also a story of the breathtaking decline of the Scottish National Party.Long the dominant player in Scottish politics, the S.N.P. has been brought low by scandal, infighting, and voter fatigue. Its formidable leader, Nicola Sturgeon, resigned in February and was later arrested by police in an investigation of the party’s finances (she was released and has not been charged).Children playing in Blantyre, as Labour Party supporters canvas the neighborhood in a campaign to win a Parliamentary seat this fall. Emily Macinnes for The New York TimesThe S.N.P.’s new leader, Humza Yousaf, has stumbled out of the gate, proving unpopular with voters, who have not rewarded him with the honeymoon in the polls that most new leaders get.Like the Tories, the Scottish nationalists, who have controlled Scotland’s devolved parliament since 2007, appear exhausted and internally divided. Their political north star — Scottish independence — seems more distant than ever after Britain’s Supreme Court ruled that the Scots cannot vote unilaterally to hold another referendum after voting against independence in 2014.While support for independence has stayed stable at around 47 percent, polls suggest it will no longer translate reliably into votes for the nationalist party. On a blustery, showery day, people in Rutherglen and the neighboring town of Blantyre said they worried more about the high cost of food and fuel, and long waiting times at hospitals — neither of which, they said, the S.N.P. government had remedied.“For me, independence takes a total back seat at the moment,” said James Dunsmore, 47, who was waiting for a haircut. The manager of the barbershop, Jewar Ali, said business had slowed because several of his cash strapped regulars were putting off haircuts to once a month.Jewar Ali, the barbershop manager, said business had slowed because several of his cash strapped regulars were putting off haircuts. Emily Macinnes for The New York TimesElizabeth Clark, 68, a retired nurse, expressed outrage at a recent newspaper report, based on credit-card receipts obtained and leaked by Labour officials, that said Scottish government officials spent public money on nail polish and yoga classes.“The S.N.P. has brought Scotland to its knees,” Ms. Clark said, her mood scarcely brightened by the flowers in her shopping cart.Feelings toward Ms. Ferrier are even more raw. After traveling by train despite testing positive for Covid — a breach of lockdown rules — in October 2020, she was suspended by the party but fought bitterly to hold on to her seat. The episode was especially embarrassing to the S.N.P. because Ms. Sturgeon had been widely praised for taking a more cautious approach to Covid than Boris Johnson did in England.“Other people were prosecuted” for breaking Covid rules in Britain, John Brown, 75, a mechanic, said over a breakfast sausage in Blantyre.“The S.N.P. has brought Scotland to its knees,” Elizabeth Clark, 68, said. Emily Macinnes for The New York TimesIn fact, Ms. Ferrier was charged with reckless conduct and sentenced to community service. After giving up her seat, she said: “I have always put my job and my constituents first, and I am disappointed that this will now come to an end.”In 2019, Ms. Ferrier was part of a wave of S.N.P. lawmakers who together won 48 seats in London’s parliament, while Labour won just one Scottish seat — Mr. Murray’s. Polls now show that the parties are virtually tied among voters, underscoring the dramatic collapse in support for the nationalist party, with the Conservatives trailing far behind. A poll last week projected that Labour was on track to win 24 seats next year, the same as the S.N.P.“It’s long been argued that unless the Labour Party can gain seats in Scotland, it will have a problem putting together a clear majority,” said John Curtice, a professor at the University of Strathclyde and one of Britain’s foremost pollsters. “It potentially significantly improves Keir Starmer’s chances of getting an outright majority.”He explained the math: With the S.N.P. maintaining its current number of seats in Parliament, Labour would need to beat the Tories by 12 percentage points just to eke out a single-seat majority (it is currently ahead by about 18 points, but Professor Curtice said that was likely to shrink). For every 12 seats that Labour wins in Scotland, it can give up two percentage points to the Tories and still gain a majority.Jackie Baillie, center, Scottish Labour party deputy leader, was among those knocking on doors on a recent afternoon. Emily Macinnes for The New York TimesGiven the peculiar circumstances of this by-election, it is Labour, not the S.N.P., that is feeling the pressure. The district has changed hands regularly since it was created in 2005; Labour won it in 2017 under the polarizing leadership of Jeremy Corbyn.“In a by-election, you’d expect the government of the day to get a kicking,” said Nicola McEwen, a professor of public policy at the University of Glasgow. “If they don’t win this seat, Starmer has bigger problems than he thinks he has.”Labour has left little to chance, mobilizing canvassers to carpet the district with leaflets for its candidate, Michael Shanks. Jackie Baillie, the party’s deputy leader, was among those knocking on doors on a recent afternoon. She played up Mr. Shanks’ roots in the community as a schoolteacher. But party officials did not make him available for an interview, suggesting they are protecting their lead.S.N.P.’s campaign office for candidate Katy Loudon, in Rutherglen.Emily Macinnes for The New York Times“It’s clearly been a difficult few months for us,” Ms. Loudon said.Emily Macinnes for The New York TimesFor the S.N.P.’s candidate, Katy Loudon, standing on doorsteps means getting the occasional tough question about Margaret Ferrier or Nicola Sturgeon. She insisted it happens less than one might expect.“It’s clearly been a difficult few months for us,” Ms. Loudon said. “But we’re in this to win. Our message is a positive one. It is not harking back to the past.” More