LONDON—Jolyon Maugham woke up early the morning after Britain’s European Union referendum and listened to the radio in shock: With most of the votes in, the odds favored leaving. He began to rack his brain for ways to stop it from happening.

“With a referendum, it’s a one-off event” and there is no chance to vote again, said Mr. Maugham, a tax lawyer in London. “The world seemed to change overnight.”

Half a year later, after consulting with academics across Europe, Mr. Maugham on Friday filed a legal challenge in Ireland’s High Court over whether lawmakers would be able to revoke Article 50—which starts the formal process of leaving the EU—after it is triggered. In two days of fundraising in December, he raised roughly £70,000 ($85,000) from nearly 2,000 people to cover the legal costs.

With lawmakers divided in Parliament, Mr. Maugham and a handful of others see the courts as their best chance at disrupting or derailing Prime Minister Theresa May’s plans for Britain’s exit from the EU. Although political analysts and economists say the cases are unlikely to be more than speed bumps, they are a rallying point for frustrated opponents hoping to maintain closer ties.

Mr. Maugham’s challenge comes days after the U.K. Supreme Court ruled in a separate case that Mrs. May must get parliamentary approval from lawmakers before starting the formal Brexit process.

The prime minister wants the U.K. to leave the bloc’s single market for goods and services, a move she says will allow Britain to abandon a requirement that lets EU citizens live in the country freely and to strike new free-trade deals, including with the bloc. Last week’s judgment potentially complicates the path out of the EU by giving lawmakers an opening to influence her stance.

The cases illustrate the clash of world views in a polarized Britain. Brexit backers call the legal efforts an attempt to undermine democracy by a liberal London elite and say Britain will have more freedom to pursue its own policies once outside the EU.

They point out that the U.K.’s economy has grown robustly since the Brexit referendum in June. Mrs. May still has approval ratings well above other prominent politicians.

But Mr. Maugham said he worries Britain will be worse off without access to the EU single market. He said voters were misled by promises about the supposed benefits of leaving, including that the national health-care system would be better funded.

“Our political class has abandoned 48% of the population [who voted against Brexit] and people are pretty desperate,” said Mr. Maugham, 45 years old, who was born in London and supported himself through the last two years of high school in New Zealand as a cleaner. “For lots of people in London, it felt like a loss of the things that they love about the United Kingdom: its inclusiveness, its liberality.”

On a personal level, he said he spent a year studying in Belgium through the EU’s student-exchange program and feels sorry that his three daughters won’t have that experience. “I saw the lives of my children getting worse in consequence of the vote,” he said.

In another case, Peter Wilding, the director of the British Influence think tank, and Adrian Yalland, a Conservative lobbyist, filed a legal challenge to the assumption that departing the EU means leaving the single market. The case is continuing.

Edward Leigh, a Conservative lawmaker who voted to leave, said there is no way U.K. courts can stop Brexit from happening or force a second referendum. He says Parliament shouldn’t have the right to overturn a democratic vote and believes Britain will do better outside the bloc, arguing that the U.K. has been hamstrung by European bureaucracy and should stand on its own.

“We’ve had a referendum, we’re going to leave, and that is that,” he said.

Stephen Booth, policy director at Open Europe, a think tank that studies Europe and took no position on Brexit, said he sees a narrow path for any impact on the course of Britain’s departure, given the results of the vote and that plans are already in motion.

“For the people behind these cases, it seems to be an attempt to make the process of leaving as difficult as possible and to throw as many hurdles in the way,” he said.

A U.K. government spokesman declined to comment on Mr. Maugham’s case, but said the government was focused on taking the U.K. out of the EU. He said the government rejects the arguments made in Mr. Wilding and Mr. Yalland’s case.

“As a priority we will pursue a bold and ambitious free-trade agreement with the European Union,” the spokesman said.

Last week, the government said the Supreme Court verdict wouldn’t affect Mrs. May’s plans to trigger Article 50 of the Lisbon Treaty, which opens the two-year window for exit talks, by the end of March. The Labour Party has said it won’t use the verdict to block Brexit, but that it would try to use the legislative process to influence the shape of a deal with Brussels.

Mr. Maugham is plying what he acknowledges is a complex case. He is arguing in a key point that Mrs. May essentially already triggered Article 50 in October when she informed her European counterparts that the U.K. was leaving the bloc, meaning, he says, that Britons have been deprived of their rights because EU countries aren’t yet willing to negotiate. He is joined as a plaintiff in the case by several members of Britain’s Green Party, which has one lawmaker in Parliament.

While lawyers for the U.K. government have said that, once started, the process of leaving the EU is irrevocable, some EU leaders say Britain could decide to reverse the process.

Mr. Maugham is hoping the case will be referred to the European Court of Justice—whose jurisdiction Mrs. May wants to leave—and he filed in Ireland because the case has to go through an EU court. If his case is successful, he hopes it will enable lawmakers to call off Brexit if, in two years, they aren’t happy with the deal—without requiring consent from all other 27 EU members.