BRITISH PRIME Minister Theresa May, who pushed to become the first foreign leader to meet with President Trump this week, appears to be hoping that a free-trade deal with the United States will ease her country’s exit from the European Union and set the stage for the new “global Britain” she envisions. Mr. Trump seems to think that reinforcing “the special relationship” with London will substitute for nurturing Western institutions such as NATO. If so, both are deluded.

Ms. May, who took power following Britain’s Brexit vote last June, shares with Mr. Trump an aversion to some aspects of globalization and a determination to limit immigration. But in a speech last week, she outlined a post-E.U. Britain that would embrace economic liberalization and aggressively pursue free trade with the rest of the world. A logical starting point, once the country forges a new relationship with Brussels, would be a free-trade pact with the United States, with which Britain already does $180 billion in business annually.

Mr. Trump assured the Times of London in a recent interview that he wanted to conclude such a deal “very quickly.” But as with many things on which the new president has promised speedy action, the words understate the hurdles. London cannot sign a trade deal with another country before it completes its exit from the E.U., which will take at least two years. And striking a bargain would require Mr. Trump to set aside the “America First” ideology he articulated last week. For example, as part of Brexit, Britain may lose the duty-free access to the rest of Europe that its car manufacturers now enjoy. Will Mr. Trump be ready to bail out Ms. May by liberalizing U.S. imports of English-built Nissans and Toyotas?

In reality, Ms. May’s embrace of a “hard” Brexit that would give up the E.U.’s single market and customs union is fundamentally at odds with a “global Britain.” Her determination to control immigration will make it difficult for potential foreign investors to recruit necessary talent and probably will prevent free-trade deals with key nations such as India. As it is, the prospect of losing automatic access to other European nations has caused big international banks to announce plans to eliminate thousands of jobs in London — hardly a step toward a European Singapore.

Mr. Trump’s notion that investing in relations with Britain and a handful of other countries — Israel, Egypt and perhaps Russia — will substitute for the web of alliances the United States forged after World War II is similarly shallow. Thanks to large defense cuts in recent years, Britain’s military will be unable to provide major support to any military operations the Trump administration launches; it certainly could not fill gaps that would be left by a breach with NATO. Just as Mr. Trump is unlikely to welcome British manufacturing imports, Ms. May does not favor a weakening of NATO or further E.U. disintegration.

Certainly, a U.S.-Britain free-trade treaty could benefit both countries and ought to be explored. But an attempt by either leader to turn the “special relationship” into an instrument for devaluing other Western alliances would damage both countries. At Friday’s summit meeting, Ms. May instead should nudge Mr. Trump toward a more positive approach to NATO and other Western institutions.