It was a carefully staged display of big-power theatrics — or more precisely, of middle-power-meets-big-power theatrics.

“You are now connected to Air Force One,” said a White House operator, as Prime Minister Keir Starmer of Britain hunched over a speakerphone, in a short video released on Saturday by 10 Downing Street.

“Mr. Prime Minister, congratulations,” said President Biden, who was flying to Wisconsin for a campaign rally. “What a hell of a victory!”

The two leaders spoke warmly about the importance of the “special relationship” between Britain and the United States, their shared commitment to defending Ukraine and an upcoming rendezvous: Mr. Starmer is leaving on Tuesday for Washington, where Mr. Biden is hosting a NATO summit.

Not since Barack Obama’s first term have the White House and Downing Street both been in the hands of center-left parties. After years of tensions over Brexit, Northern Ireland and frustrated British hopes for a trade deal, this could augur a new era of harmony in the trans-Atlantic relationship.

It could also be a brief era. The American election in four months could restore to the presidency Donald J. Trump, who managed to have vexed relationships with a right-of-center British leader, Prime Minister Theresa May, let alone a left-of-center one.