If Britons vote to exit the European Union — and the latest polls show suggest that’s a growing possibility — who would have the last laugh on this side of the pond? Well, you read it here first*: Donald Trump.

That is starting to dawn on the British establishment, to judge by the coverage this week in the London Financial Times. It just trotted out the ex-president of Harvard, Lawrence Summers, to warn that “Brexit” and Mr. Trump have “much in common.”

As President Summers sees it, both the Brexit and Trump campaigns pit “populists against the political establishment” and, he reckons, both are “angry.” And in both cases, “polling suggests the outcome is in doubt.”

The latest poll, by Britain’s Independent Television Network, found that for the first time, the “Leave” camp is outpolling those who would “Remain” within the EU. That poll puts the margin at four points, 45% to 41%.

So hot is the question that the government had to scramble this week to extend the deadline for registering to vote in the referendum. That’s because a surge in late voter registrations had crashed the online system.

The surge followed the release on Monday of another poll, by TNS, suggesting the “Leave” camp is ahead of the “Remain” camp by 43% to 41%. It was the first time the sentiment for Brexit had moved into the lead.

It’s always possible the polls will reverse themselves or prove inaccurate. The bookies in Britain are giving longer odds on to the exit than they are giving on the status quo. (Two years ago, though the polls suggested it would be close, the Scots shrank from voting for independence from the United Kingdom.)

If the tide has turned for Brexit, my own theory is that it began when President Obama showed up in London to tell the Brits to vote to remain in EU. It was a shocking act of interference in the internal affairs of an ally.

Not only did Mr. Obama urge Britons to vote to stay within Europe. He also warned them that if they voted to leave, they’d have to get to the “back of the queue” before they could get a trade deal with America.

Hillary Clinton fell right in behind him, ignoring the recognition in Britain that Obama had made a threat. She put out a statement to the London Observer saying that she “has always valued a strong United Kingdom in a strong EU.”

Which means that the former secretary of state, who is basing her campaign largely on her foreign policy chops, seems to be out of touch with our oldest ally. While her supposedly inept opponent has, in Mr. Trump, been far more adroit.

Mr. Trump, an investor in Britain, told Fox News the other day that he supported a British exit from the European Union, which puts its member countries under control of statist bureaucrats based in Brussels and has an anti-American streak.

The Donald, in contrast to Obama, was careful to avoid trying to influence the vote per se. “I would say that they’re better off without it, personally,” he said in reference to the European Union, but added: “I’m not making that as a recommendation.”

Is it possible to imagine — that in the real world of serious international issues, Donald Trump could be more diplomatic than Hillary Clinton? Or would be more tolerant of democratic processes by our allies?

It’s an important question. If the Brits do vote to stand for their independence from Europe, after all, the betting is that David Cameron’s goose will be cooked and the Conservatives will be looking for a new prime minister.

That could — or could not; it’s a stretch — bring to 10 Downing Street the former mayor of London, Boris Johnson, who is similar to Mr. Trump in that he’s a populist but different in that he’s a bicycle-riding ex-newspaperman with a profound knowledge of history. Then again, too, they both have unruly heads of hair. What a special relationship they could craft.

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* Conrad Black, a reader notes, has been all over this story, including in his column of May 16.