LONDON—Britain’s Labour Party has won a landslide election victory as voters look set to hand its leader Keir Starmer one of the biggest parliamentary majorities in British history and place a center-left government into Downing Street for the first time in 14 years.

Projections by the BBC predicted that Labour will win 410 of the 650 seats in Parliament. The ruling Conservative Party is on course to win just 144 seats, likely the worst result in its 190-year history and a massive reversal on its victory in the last election in 2019.

Prime Minister Rishi Sunak early Friday conceded the election and apologized to his party for the scale of the defeat. “The British people have delivered a sobering verdict tonight…and I take responsibility for the loss,” he said.

“Change begins right here,” said Starmer, after retaining his seat.

Smaller parties performed strongly, showing voter discontent with mainstream politicians. Populist Nigel Farage, who leads the anti-immigration Reform UK party, made it into Parliament for the first time along with several other leading party figures. Reform’s gains denied the Conservatives dozens of seats across the country by splitting the right-wing vote, handing more seats to Labour.

Thursday’s election marked a huge shift in Britain’s political landscape and a heavy repudiation of Sunak’s Conservative Party, which has been in power during a tumultuous decade in British politics that included Britain’s departure from the European Union , political infighting and scandals that saw four prime ministers in five years, the pandemic, war in Ukraine and a cost-of-living crisis.

Sunak, Britain’s first nonwhite leader, became prime minister 20 months ago after his predecessor Liz Truss rolled out huge tax cuts that spooked financial markets and sparked a run on the pound, forcing her to quit just weeks into her term.

Starmer, a former public prosecutor, has pledged to bring a period of pragmatic calm to British politics and to “stop the chaos” of the Conservatives.

“It’s a happy day,” said Tom Gunner, a 59-year-old Labour supporter. Gunner, who was celebrating his birthday and baked and brought cookies to a polling station in north London, said he’s keeping his expectations in check, however. Starmer “inherited a real mess,” he said.

Thursday’s vote is likely to be the latest example of growing voter frustration with incumbent political parties across many democracies, when the economic fallout from the pandemic and war in Ukraine have sparked high inflation and damaged incomes. From South Africa to India to France, many voters are punishing politicians in power.

The scale of the Labour victory—a majority of about 170 in Parliament—suggested by BBC projections means that parts of Britain that have voted Conservative for generations will likely switch to Labour, handing it a huge mandate to begin patching up a nation that has seen its economy stagnate and public services such as hospitals and railroads degrade.

The BBC projection also suggested that Reform UK would win four seats. Farage has said he would use his seat in Parliament to try to draw like-minded lawmakers from what is left of the Conservative Party and form a new right-wing voting bloc.

“Believe me folks, this is just the first step of something that is going to stun all of you,” said Farage in his victory speech as he was elected to Parliament in his eighth attempt.

While Reform got far fewer seats than the Tories, pre-election polls showed it could capture about 15-17% of the popular vote, only a few points shy of the Tories at about 20-22%.

“Much of the damage to the Conservative Party tonight is being done by Reform, even if Labour is the beneficiary of that,” said John Curtice , a professor of politics at the University of Strathclyde.

Since the last election in 2019, Britain has been bruised by the disruption of Brexit, its divorce from the EU which took effect in early 2020 and further slowed economic growth. The scars are still visible. The state-run public-health system has 6.3 million people waiting for treatment. Some 2.8 million people are off work sick.

For the first time since records began in 1955, British households are, when adjusted for inflation, on average poorer after a parliamentary term, according to the Resolution Foundation, a London think tank. Brexit , meanwhile, has proved a disappointment to many of its supporters. It ended the free movement of other Europeans allowed to move to the U.K. for work, but immigration rose anyway to record highs in 2022 and 2023 and is only slowly falling back.

Unlike some countries in Europe such as France and Germany that are seeing the sustained rise of far-right parties, Britain will tilt to the left. Not that people may notice a major shift: Starmer has moved the party sharply toward the center in recent years, shedding its more radical policies and members, and has promised to continue Britain’s pro-U.S. foreign policy, including continued support for Ukraine and Israel.

Starmer’s predecessor as Labour leader, the socialist Jeremy Corbyn , ran this time as an independent candidate and retained his long-held seat in London.

While many voters in the West are leaning toward charismatic figures to solve their problems, the U.K. is now poised to have a technocratic Labour leader who is widely seen as a dry pragmatist and hasn’t made any big promises other than to run a more efficient and honest government.

Despite the record victory, Starmer’s approval ratings are negative in many polls, as trust in politicians more widely sits at record lows. A YouGov poll showed 48% of those who planned to vote for Labour said it was to get rid of the Conservatives. The next most popular answer was to make way for change at 13%. Just 5% said it was because of Labour’s policies.

“I kind of voted for a party that I’m not wholly into,” said David McGing, 25, who voted Labour in south London. He was also considering voting for the smaller Green Party but didn’t want to take a risk and wanted the Tories out.

For the Conservatives, widely seen as the world’s most successful political party in the past 150 years in terms of years in power, a loss on this scale is a huge reversal of fortunes. Just five years ago it won an 80-seat majority on a promise to get Brexit done. The issue of Brexit allowed the Conservatives to build a new coalition stretching from the postindustrial heartlands who felt left behind by globalization to the party’s traditional aspirational middle-class supporters.

But that coalition has come under strain in the years since. Some working-class voters who supported the Tories in 2019 are switching back to Labour, while others went to Reform. Districts considered Tory heartlands are going to Labour. Sunak, the lawmaker for a district in Yorkshire, is even battling to hold on to his own seat.

The loss threatened to eclipse the Tories’ 1906 result of just 156 seats.

The general disaffection means a handful of smaller parties are on track to perform well as they siphon protest votes. The Liberal Democrats, whose leader Ed Davey has gained headlines by being filmed during his campaign doing publicity stunts such as bungee jumping and falling into rivers, could record one of their best performances, with 61 seats predicted by the BBC.

Reform also did well on a message to cut immigration. The Conservatives have long promised to reduce immigration, and some Britons who backed Brexit did so in the hopes that it might reduce the numbers of workers entering from Eastern Europe. But instead, immigration has risen to record levels to ease labor shortages, with legal migrants from Africa and India largely displacing immigration from Eastern Europe. That left many who backed Brexit feeling disillusioned.

“I think the British public, now, is fed up with lies, deceit, corruption, the establishment,” said Peter Lee, a 73-year-old in southern England. Lee said he planned to vote for Reform to see if the party can deliver lower immigration, which reached record levels under the Conservatives, despite their pledge to bring it down. “Whether it’s to do with Brexit, Covid, the Ukraine war. It’s just lies after lies,” Lee said.

A big challenge for Starmer and Labour is they won’t have much money to spend to improve public services such as the healthcare system and an aging network of railroads. The last time Labour won big was in 1997 when Tony Blair ended 18 years of Tory rule amid a wave of enthusiasm. But Britain’s economy was growing much faster in the late 1990s and 2000s, and growing tax revenues allowed Blair to increase spending on everything from schooling to healthcare.

This time around, there is little room to maneuver. To shield the economy from the pandemic and the war, the British government’s tax take as a percentage of the economy is at its highest level since World War II, and government debt has climbed to 90% of annual economic output. A slow-growing economy, meanwhile, isn’t providing extra tax revenues.

“You’re stuck with a cake that isn’t growing, so how do you cut up the cake to give people more of what they want?” said Tony Travers, a professor of politics at the London School of Economics.

Starmer, trying to shed Labour’s image as a party that taxes and spends too much, has offered only targeted tax increases to patch up the nation’s public services and is pledging to keep government debt in check. He and the likely next Treasury chief, Rachel Reeves , are planning to cut red tape to build more houses; reduce immigration; create a fund to accelerate the build-out of green-energy infrastructure; and make it easier for people to get appointments in the health system.

They have also had to scale back their plans. Labour abandoned its more ambitious idea to borrow £28 billion, equivalent to $35.4 billion, to fund a British equivalent to the U.S. Inflation Reduction Act last summer as government borrowing costs remained high.

Write to Max Colchester at Max.Colchester@wsj.com and David Luhnow at david.luhnow@wsj.com