Voters went to the polls in Britain on Thursday in a dyspeptic mood, many of them frustrated with the Conservative government and skeptical that any replacement can unravel the tangle of problems hobbling the country.

Their skepticism is warranted, according to analysts. Even if the Labour Party wins a robust majority in Parliament, as polls suggest, it will confront a raft of challenges, from a torpid economy to a corroded National Health Service, without having many tools to fix them.

The Labour leader, Keir Starmer, would inherit a “legacy of ashes,” said Robert Ford, a professor of political science at the University of Manchester. And voters, who less than five years ago elected the Conservatives in a landslide, are not likely to give Mr. Starmer much slack to turn things around.

“The message could not be clearer: You must deliver change — or you are toast,” Mr. Ford said. “People will not be patient.”

The election is shaping up as a political watershed for the country. It is likely to represent the repudiation of the Conservative Party after 14 years in power, and the elevation of the Labour Party, which less than five years ago suffered its own electoral defeat, the worst since 1935, at the hands of the Conservatives.

That dizzying reversal of political fortunes speaks to the turmoil that has engulfed Britain since it voted to leave the European Union in 2016. Brexit fractured the Conservative Party, leaving it increasingly erratic and extreme, as it confronted the challenges of the coronavirus pandemic and a cost-of-living crisis.

With Labour on the threshold of power, it faces hard trade-offs that reflect the fallout from those years: a country depleted, exhausted and desperate for change.

Rebuilding the Economy

Labour’s central sales pitch is that it can turbocharge the economy and generate enough extra revenue from taxes to avoid deep cuts in public services, tax increases, or greater borrowing. But its levers for doing so are limited, particularly after its ambitious multi-billion-pound plan to transform Britain into a “green economy” fell victim earlier this year to the government’s weak financial position.

Another option would be to ease restrictions on trade with the European Union, which have hampered British exporters since Brexit. Yet Mr. Starmer has ruled out rejoining the bloc’s vast single economic market, since that would mean allowing people from Europe the freedom to live and work in Britain, or its Customs Union, which would mean accepting some of the bloc’s rules on tariffs and duties.

While analysts said it should be possible for a Labour government to strike more limited deals, like a new trade pact on animal and plants that would help British food exporters, these would provide only a small boost to the economy.

That leaves a lot riding on another major Labour goal: overhauling Britain’s planning system to build more houses and speed up construction of public works projects. Labour is expected to review which parts of the countryside remain off limits to developers and restore targets for residential building in urban areas.

Rewriting the planning rules could unlock a building boom, economists said, but it would upset voters who want to protect green spaces.

Budget pressures would also complicate Labour’s efforts to fix the N.H.S., where the government failed to cut waiting times that stretch to months. Labour has promised to schedule 40,000 additional appointments a week, which it claims it can do by persuading N.H.S. workers to take on more appointments during off hours.

Labour said it would also hire 8,500 health workers to treat patients with mental health issues, as well as doubling the number of C.T. scanners and M.R.I. machines in hospitals. It would finance this by closing a loophole for people who claim non-domiciled status in Britain, allowing them to avoid paying some taxes.

But its immediate challenge would be to resolve a long-festering wage dispute with junior doctors. After the government rejected their demands for a 35 percent wage increase, the doctors have repeatedly walked off the job, causing a cascade of canceled procedures and prolonging waiting times.

Addressing Immigration

Tackling immigration is another challenge for Labour, not least because of the labor shortages at the N.H.S. and throughout the economy. Controlling Britain’s borders was a big theme of the Brexit referendum, but net legal migration — the number of people who arrived, minus those who left — has roughly tripled since then, hitting ‌almost 750,000 in 2022, before falling back slightly.

Labour will benefit from changes introduced by the Conservative government, which restricted the right of students to bring relatives to Britain, and by the reduction in the number of refugees coming from Ukraine, Hong Kong and Afghanistan. All told, net migration is now expected to fall.

But against that backdrop, Labour would have to work hard to fulfill its promise to train more people from Britain to fill job openings, and to dissuade employers from looking abroad for workers.

Curbing the flow of asylum seekers landing in small boats on the English coastline would be harder still. Mr. Starmer has vowed to scrap a costly policy under which some asylum seekers would be put on one-way flights to Rwanda. Labour would instead try to crack down on people-smuggling gangs, while bolstering cooperation with authorities across continental Europe.

Yet Britain has already given tens of millions of pounds to the French to help them stop the small boats, with only partial success. It also remains unclear how much scope there will be for better collaboration, with far-right, anti-immigration parties making striking gains in elections underway in France.

At home, Britain faces a backlog of asylum applications, and the cost of housing some of those awaiting decisions, many in hotels, is around 8 million pounds, or $10.2 million, a day. Labour has promised to hire 1,000 new caseworkers to help remove those whose applications fail. But many are from countries that have no agreement with Britain to accept failed asylum seekers.

Facing an Uncertain Ally

Any British leader would face an increasingly murky political landscape in the United States. The questions about President Biden’s viability as a candidate in the upcoming election have increased the odds that his Republican opponent, Donald J. Trump, will recapture the presidency.

Labour’s top foreign policy official, David Lammy, has tried to cultivate people in Mr. Trump’s orbit, including Senator J.D. Vance, Republican of Ohio. But Mr. Lammy’s calling card in the United States is his close relationship with former President Barack Obama. The two men went to Harvard Law School, and Mr. Lammy campaigned for Mr. Obama during his first presidential campaign.

Mr. Starmer’s ties to the United States are not as deep. While he does not have a history of critical remarks about Mr. Trump, there is little to suggest that a 61-year-old former chief prosecutor would develop a strong relationship with 78-year-old man who is a defendant in multiple criminal cases.

On Wednesday, however, Mr. Starmer won backing from another unlikely source: Rupert Murdoch. His influential London tabloid, The Sun, endorsed Labour for the first time in an election since 2005.

“Time for a New Manager,” the Sun said on its front page, playing off the European soccer championship, in which England’s national team has struggled but remains in contention as it enters the quarterfinals.

“By dragging his party back to the center ground of British politics for first time since Tony Blair was in No. 10,” the paper said, “Sir Keir has won the right to take charge.”