Think again? Calls mount for British to vote again on Brexit.

On a busy market square, Diane Holden stands by a whiteboard attached to a metal railing. “Brexitometer” it reads – a play on Brexit, the all-consuming national drama – above a row of columns.

Ms. Holden and her Brexitometer are part of a widening campaign to hold a second referendum on whether Britain should leave the European Union, which is what 52 percent of voters backed in a 2016 vote. Campaigners want another plebiscite on the Brexit deal, a do-over that could stop it in its tracks.

“How do you feel about the process?” she asks shoppers, competing with the throaty cries of fruit-and-vegetable sellers for their attention. “Do you think it’s going well?”

Those who stop to talk are invited to put colored dots in the whiteboard columns. Aylesbury, a prosperous rural market town 45 miles northwest of London, narrowly voted “Leave” in 2016, but many residents are unhappy with the way things have worked out since. Is Brexit going well? (Most chose No.) Will it be good for jobs? (No.) Should there be a new national vote on the final deal? (The clincher: Yes.)

What began as a quixotic and quarrelsome campaign, dismissed by Brexit backers as a fringe rearguard action, has in recent months moved into Britain’s political mainstream. Last month 700,000 people marched in London in favor of a “People’s Vote,” and support has built up among members of parliament from all parties.

Between ‘vassalage’ and ‘chaos’?

Last week Prime Minister Theresa May unveiled her draft agreement with the EU on the terms of Britain’s withdrawal. It sparked a storm of criticism within her ruling Conservative Party, largely because it would keep Britain in the EU customs union for several years at the cost of limiting national autonomy. Should parliament reject the deal, as seems likely, Britain would be on track to crash out of the EU in disorderly fashion, with no deal in place.

That stark prospect has begun to rattle politicians who had backed Ms. May’s Brexit strategy. Jo Johnson, whose brother Boris Johnson is a prominent Brexiter, quit earlier this month as her transport minister. What Britain was being offered, he complained, was a choice between “vassalage” (May’s deal) and “chaos” (no deal). He called for a second referendum.  

In an editorial, the left-leaning Observer newspaper wrote on Sunday that “There is a watertight case for a referendum on the deal: we now know the terms of exit, there are huge unforeseen costs, and the public must have a say.”

Still, the political and legal paths to voting again on Brexit – and potentially reversing it – are tortuous. Both May and her opposition counterpart, Labour leader Jeremy Corbyn, oppose the idea. There is little time left to dither: Britain is due to leave the EU on March 29 next year. For many voters who ticked the “Leave” box in 2016, another referendum would be a betrayal of what had been proclaimed a binding vote.

Public opinion remains sharply divided on Brexit, though polls now show an edge for “Remain,” says John Curtice, a professor of politics at Strathclyde University who closely tracks polling data. That edge, however, is largely down to the views of young people, who turned out in lower numbers than their elders in 2016, and of those who abstained.

Even if Britain were headed for a cliff-edge, crash-out exit, “do not expect Leave voters to change their minds” in a second vote, says Professor Curtice.

A narrow majority also supports a second referendum, widely seen as a bid by Remainers to have a second bite at the cherry they missed last time.

A threat to democracy?

In the wintry sunshine of Aylesbury there is plenty of grumbling at the government’s handling of Brexit and trepidation at what will follow. But there is little sign of opinion shifting at this defining political moment for Britain.

Lorraine Bail, whose husband works in Belgium, voted Remain in 2016. But she’s not sure that holding another referendum is the answer to a political impasse. “We voted, and even though I disagree with the vote, we’re in a democracy and the majority has to have their say,” she says.

“It seems to be taking so long,” says Bob Dormer, a driver who abstained in 2016. At the garage where he works, Leave was a popular choice. He puts this down to the promise of more money for public health services, which has since been debunked, and says he would like to see another referendum.

“A lot of people voted to go out and now realize what a waste of money it’s been,” he says.

Holden was one of half a dozen activists on Saturday at the Brexitometer, handing out stickers and leaflets to passersby. They did not impress Philip Skola, a retiree, who stopped to berate one of the volunteers. “If you respect democracy, we voted to come out and that’s how it should be,” he said.

Asked whether it would not be fairer to give people a say on the final terms of the deal, Mr. Skola’s face darkened. “If we lost this then I’d never vote again. Democracy is dead,” he snorted.

Analysts have warned of civil unrest if the results of the first referendum are overturned and some say a second campaign would be more divisive than the last one. It’s not clear what question would be on the ballot; a three-way choice between the terms of May’s deal, crashing out with no deal, or simply remaining in the EU, might yield no decisive outcome.

Bad math in Parliament

For the time being, the question is moot; Parliament must vote first. But the math there looks bad for May, and “if the government loses the vote in the House of Commons, all bets are off,” Curtice says.

If Britons are indeed asked again to make up their minds on EU membership, the future is unclear. It would not take much of a shift to change the last referendum’s outcome, and while most voters have stuck to their 2016 views, some have changed their mind.

Emma Jiao-Knuckey is one of them. She lives in Southend-on-Sea, a strongly Leave constituency, and she voted with the majority in 2016, persuaded by the prospect of more money for health and social spending. Her two sons are both diagnosed as autistic.

Leavers based much of their appeal on a promise that if London was no longer paying in to the EU budget the government would have $450 million a week to spend on its creaking National Health Service. This was untrue.

Now Ms. Jiao-Knuckey is a volunteer for the People’s Vote campaign and joined its recent march in London. The 2016 referendum result does not constitute informed consent, she says, and it would be better if voters had a say on the final terms of the withdrawal deal.

However the vote turned out, she says, “at least I’d have the peace of mind of knowing that we were informed. We’ve got to live with this for a very long time.”