LONDON— Rishi Sunak, a former hedge-fund manager and U.K. Treasury chief, formally became Britain’s prime minister on Tuesday after he was appointed by King Charles III, vowing to steer the U.K. through a period of growing political and economic troubles.

In his first speech as prime minister, Mr. Sunak warned of a “profound economic crisis” facing the country, which is suffering from stagflation and has recently plunged into deeper political uncertainty with three different prime ministers in seven weeks.

Mr. Sunak said he would keep the current chancellor of the Exchequer, Jeremy Hunt, who stepped in to undo Ms. Truss’s tax-cut plan and regain market confidence.

Britain’s first Hindu leader pledged to repair the damage caused by the ill-fated experiment in British Reaganomics of his predecessor, Liz Truss, who was forced from office after markets were spooked by large unfunded tax cuts and a generous subsidy for household energy costs.

“Some mistakes were made. Not born of ill will or bad intentions. Quite the opposite in fact, but mistakes nonetheless. And I have been elected as leader of my party and your prime minister, in part to fix them,” said Mr. Sunak, as he stood in front of Downing Street. “I will place economic stability and confidence at the heart of this government’s agenda.”

King Charles III welcoming Britain's new prime minister, Rishi Sunak, at Buckingham Palace on Tuesday.

Photo: POOL/via REUTERS

Mr. Sunak takes control of a Conservative Party that has its lowest rating in the polls in decades. He will have to orchestrate one of the great political rebrands of recent British political history if he is to lead them to a victory during an election expected in 2024, analysts say.

Mr. Sunak moved quickly Tuesday to steady the Conservative Party. He appointed lawmakers from across warring factions to senior government roles in an effort to rebuild some unity in a party that has ousted its past three prime ministers in as many years. Nearly all the top appointments were cabinet members of former Prime Minister Boris Johnson, who was pushed to resign in July by a party rebellion.

In a sign of the challenges facing Mr. Sunak, the new prime minister broke with tradition and didn’t have allies in Downing Street clapping him into the building, underscoring the dark economic times the nation faced as he prepares to oversee some difficult decisions to plug a government budget deficit that is estimated to be 40 billion pounds, equivalent to $45 billion.

“I will unite our country not with words, but with action,” Mr. Sunak said. “I will work day in and day out to deliver for you.”

President Biden spoke Tuesday with Mr. Sunak to congratulate him on his appointment as prime minister, according to a U.K. government readout. The two men discussed bilateral cooperation, including efforts to counter China’s “malign influence,” as well as efforts to bolster Ukraine in its war against Russia.

Mr. Sunak, who at 42 years old is Britain’s youngest leader in more than 200 years, faces a daunting inbox. The British population is struggling with a cost-of-living crisis as inflation runs at 10.1%, fueled by high energy costs because of the war in Ukraine. With financial markets now wary of the stability of U.K. government finances, Mr. Sunak will have to regain market confidence through a combination of politically damaging spending cuts and tax increases, likely aggravating a recession and hurting incomes further, analysts say.

The government is set to outline spending cuts on Monday, only days before the Bank of England is expected to also raise interest rates.

“It’s going to be a terrible time for the economy whoever is in power,” said Jill Rutter, a former government official and a senior research fellow of U.K. in a Changing Europe, a think tank. “It will be very difficult with any government to come through that with the voters saying, ‘That was great.’ ”

Investors have welcomed the end of Ms. Truss’s government and the shift in policy toward more fiscal caution. The pound has fully recovered from its selloff following the tax-cut announcement on Sept. 23, which saw sterling briefly hit a record low of $1.0349. The pound traded at $1.1480 on Tuesday, roughly 2% above its prebudget level.

U.K. government bonds, which were at the heart of the recent U.K. market turmoil, have also staged a strong rally that continued Tuesday as Mr. Sunak took office. The yield on a 10-year U.K. gilt was at 3.647% Tuesday, well below a high of 4.643% set earlier this month, according to Tullett Prebon data. Yields rise as prices fall.

“It’s helpful that we have a resolution, at least for now, to the craziness of the last few weeks,” said Fraser Lundie, head of fixed income for public markets at Federated Hermes in London. “Today and yesterday is the first time where you could start thinking in weeks instead of days. Perhaps in the weeks to come you can start thinking in months.”

But as investors start to take the longer view, they may not like what they see in the U.K. economy, he added. “As the days go on I think people will pretty quickly change their attention from that crazy crisis period back to watching the Bank of England, watching the economic picture. It doesn’t look great to be honest,” he said.

Mr. Sunak’s opening statement came just over an hour after Ms. Truss defended her vision for a low-tax, high-growth economy.

Mr. Sunak takes control of a Conservative Party that has its lowest rating in the polls in decades.

Photo: HENRY NICHOLLS/REUTERS

“As the Roman philosopher Seneca wrote: ‘It is not because things are difficult that we do not dare. It is because we do not dare that they are difficult,’ ” she said in a farewell speech outside Downing Street before handing her resignation to King Charles. “We simply cannot afford to be a low-growth country where the government takes up an increasing share of our national wealth.”

Polls this week showed that Ms. Truss had the lowest approval rating of any prime minister in modern times, with one survey giving her a 6% approval rating.

The Conservative Party “is in free fall and I don’t know if it has a parachute or not,” said Matthew Goodwin, a politics professor at the University of Kent.

In his cabinet shuffle, Mr. Sunak kept Defense Secretary Ben Wallace in his post as well as Foreign Secretary James Cleverly. Suella Braverman, who is popular on the libertarian wing of the party and advocates tough migration restrictions, was named as home secretary.

Mr. Sunak inherits a political machine that is accustomed to rapid rebrands. The Conservative Party, which was founded in 1834, is one of the world’s oldest and most successful electoral franchises. Its success lies in its ability to repeatedly shed its skin and emerge anew to appeal to the ever-changing needs of its electorate.

The past 12 years of Tory government is a prime example. Under Mr. Sunak, the party will have completed an ideological full circle that started when David Cameron came into office in 2010 as a social liberal and fiscal conservative, as his government tried to repair the nation’s finances after the 2008 financial crisis. After the Brexit vote in 2016, Theresa May tried to launch the party in a new socially conservative direction. Her successor, Mr. Johnson, remodeled it into a more populist franchise as he bulldozed through Brexit and ushered in a period of state intervention and high taxes. Ms. Truss took over and tried to rapidly dismantle that with an unsuccessful shift toward free markets and lower taxes.

Mr. Sunak is now expected to take it back to the Cameron-era focus on deficits with a degree of social liberalism, embracing issues such as climate change. It is unclear whether the electorate will buy the Conservative Party’s latest rebrand.

Ms. Rutter recalls being in government when the Conservatives successfully retooled their economic policies after the 1992 “Black Wednesday,” when speculator George Soros and other hedge funds forced the pound to break its peg to European currencies. Despite their best efforts, the Tories were never fully forgiven by voters and later spent 13 years out of office. Having worked hard to rebuild their brand as competent on the economy after 2010, the Conservatives “had an economic competence premium, and Truss managed to burn through that,” Ms. Rutter said.

In a February speech, Mr. Sunak, then chancellor, laid out his views on the challenges facing the U.K. and other Western economies where economic growth is slowing and productivity is stagnant. He warned that failure by politicians to create the conditions for faster growth would undermine public support for free-market economies and democracy in a world where autocracies such as China are on the march.

But he also warned against what he described as two false ideas on how to spur growth. The first was more government spending, regardless of its impact on borrowing and debt. The second was unfunded tax cuts, the idea that slashing taxes will unleash growth that will eventually give the government more money from a dynamic economy to spend on social services and investment. The latter idea is precisely what his predecessor, Ms. Truss, tried and failed to carry out some seven months after Mr. Sunak’s speech.

“The trap of both those ideas—that we can simply boost the economy with public spending, or supposedly self-funding tax cuts—is that they are both highly seductive, easy answers,” he said. “Neither are serious or credible; neither on their own will transform growth; and because they ignore the trade-offs inherent in economic policy, both are irresponsible.”

Instead, Mr. Sunak outlined three areas he called capital, people and ideas aimed at getting businesses to invest more. He said capital invested by British companies averages 10% of annual economic output versus a 14% average in the Organization for Economic Cooperation and Development club of rich countries. He pledged to help drive innovation by creating the right tax and regulatory environment for business to boost capital investment and spending in research and development, called for more vocational training of employees already in the workforce, and a visa system to attract entrepreneurs and high-skilled workers.

“Less ‘build it and they will come’ and more ‘let them come and build it,’” he said.

Mr. Sunak will have to navigate opposition from lawmakers within his own party to increased immigration of any kind.

On Monday, Mr. Sunak warned lawmakers during a private meeting that they had no option but to cooperate if they wanted to avoid losing the next election, according to people present. He is hoping that the Tory lawmakers’ desire for self-preservation will trump their personal ideological leanings, one person added.

Mr. Sunak inherits a healthy majority in Parliament following Mr. Johnson’s 2019 electoral victory and so should be free to push through legislation as long as he can contain rebellions.

Mr. Sunak said he would stick to the 2019 manifesto that helped Mr. Johnson secure his electoral victory. It included a pledge to help left-behind parts of the country and crack down on illegal migration. “I will deliver on its promise,” he said.

—Chelsey Dulaney and Vivian Salama contributed to this article.

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