The National Police Chiefs’ Council in Britain called the leaks a breach of trust, adding, “This damage is even greater when it involves unauthorized disclosure of potential evidence in the middle of a major counterterrorism investigation.” The disclosure of potential evidence “undermines our investigations and the confidence of victims, witnesses and their families,” it added.

On Thursday, Manchester’s top police official, Chief Constable Ian Hopkins, joined the chorus of criticism, saying that the disclosure “has caused much distress for families that are already suffering terribly with their loss.”

Earlier in the day, the BBC reported that the Manchester police would no longer share details of the investigation with American counterparts. But on Thursday evening, after Mrs. May had new assurances from Mr. Trump, the police announced that intelligence sharing had resumed — if it had ever stopped in the first place.

Mark Rowley, an assistant commissioner in charge of counterterrorism at Scotland Yard and an officer on the Police Chiefs’ Council, said in a statement issued later on Thursday that “while we do not usually comment on information-sharing arrangements with international law enforcement organizations, we want to emphasize that, having received fresh assurances, we are now working closely with our key partners around the world including all those in the ‘Five Eyes’ intelligence alliance.” (Along with the United States and Britain, the other countries in the alliance are New Zealand, Australia and Canada.)

The Times said in a statement:

The images and information presented were neither graphic nor disrespectful of victims, and consistent with the common line of reporting on weapons used in horrific crimes, as The Times and other media outlets have done following terrorist acts around the world, from Boston to Paris to Baghdad, and many places in between. Our mission is to cover news and inform our readers. We have strict guidelines on how and in what ways we cover sensitive stories. Our coverage of Monday’s horrific attack has been both comprehensive and responsible. We cover stories about terrorism from all angles. Not only stories about victims but also how terrorist groups work, their sources of funding, how they recruit. Acts of terrorism have tremendous impact on how we live, on how we are governed and how we interact as people, communities and nations. At times the process of reporting this coverage comes at personal risk to our reporters. We do it because it is core to our mission.

Mr. Trump has viewed leaks differently at different times depending on whether they helped or hurt him. During last year’s presidential campaign, he not only capitalized on the disclosure of emails from the Democratic National Committee and from Hillary Clinton’s presidential campaign, he publicly called on Russian hackers to unearth and publicize even more of them. “I love WikiLeaks,” he said at one point, praising the group that made public many of the emails.

But since taking office, Mr. Trump has been increasingly frustrated by information coming out of his own White House. Details of his conversation with Russian officials and of his telephone calls with the leaders of Mexico, Australia and, just this week, the Philippines have spilled into public view. Leaked information about a telephone call between Michael T. Flynn, his first national security adviser, and Russia’s ambassador forced the president to fire Mr. Flynn.

Mr. Trump’s own aides also routinely leak about one another in the latest palace intrigue.

In recent days, anonymously sourced articles about Mr. Trump’s private conversations with and about James B. Comey, the F.B.I. director he fired, have fueled investigations into his associates’ ties with Russia. After The Times reported that Mr. Trump had asked Mr. Comey to shut down an investigation into Mr. Flynn, the Justice Department felt compelled to appoint a special counsel to take over the Russia investigation.