British authorities have named two of the three suspects in the attack.
She said she was "intensely proud" of the police and emergency services who responded to the terror attack, and also paid tribute to courageous members of the public.
Although the spat is embarrassing, Trump's state visit - which according to traditional practice would include staying at Buckingham Palace as a guest of the queen - does not appear to be in any jeopardy.
"Clearly the people responsible for these horrific terror attacks are the bloodthirsty and depraved terrorists but Theresa May has to take responsibility for letting austerity damage her ability to keep us safe". His brash attitude had already set European teeth on edge after his foreign trip last month.
Just as news spread that a van had apparently run down pedestrians on London Bridge in the city's center, Trump-though he sits at the head of the world's most sophisticated intelligence apparatus with massive information-gathering capabilities-took the opportunity to retweet a message from Drudge, a site known for anti-Muslim bigotry and sensationalist headlines. "We will release facts when we can - our info must be accurate #LondonBridge".
Asked if Trump was criticizing the mayor of London because he is Muslim, Sanders said that was "utterly ridiculous".
The mayor has been at meetings of the Government's emergency Cobra committee and "we've been working with him to ensure the response was right, and to get London moving again", the Prime Minister added. We need the courts to give us back our rights.
He said there is also great unity and determination because "our unity and love for one another will always be stronger than hate from the extremists".
On Twitter, Mr Trump said if "we don't get smart it will only get worse".
"Whatever the United States can do to help out in London and the U. And we, as a society, especially as Americans, better do something about it". "GOD BLESS!" Trump tweeted Saturday night.
Trump to announce decision on Paris climate pact Thursday
Trump has repeatedly expressed doubts about climate change, at times calling it a hoax to weaken United States industry. President Donald Trump is expected to announce his decision on whether to abandon the Paris climate accord Thursday.
An on-camera appearance expressing support and vowing to defeat terror wherever it emerges might follow.
Staff of previous president Barack Obama were among those who tweeted their disgust. And they certainly don't seem to be laying out a plausible anti-terrorism strategy. Prime Minister Theresa May invited Trump a week after his January 20 inauguration.
Trump's first comment after the attack came late Saturday on Twitter. Trump posted that the U.S. Department of Justice should have pushed his "original" travel ban, or sought a tougher version of the executive order meant to temporarily bar citizens of six predominately Muslim countries from entering the United States, and not the "watered down, politically correct version" now in front of the Supreme Court.
Had those great presidents done such things, we would not now think of them as being so great.
Seeking to reassure London residents and visitors, Khan had told BBC Radio on Sunday that "Londoners will see an increased police presence today and over the course of the next few days".
Was he childishly indulging a personal grudge against Khan who, during the presidential election, had shown disfavor for Trump?
On Monday the president brushed aside criticism of his original posting, tweeting: "Pathetic excuse by London Mayor Sadiq Khan who had to think fast on his "no reason to be alarmed" statement. That was the point he was trying to make". "They'll say something you'd expect and then he'll go off and bring in the gun debate to a terror attack". "That's because they used knives and a truck!"
The White House on Monday denied suggestions from USA reporters that Trump "picked a fight" with Khan "because he was Muslim".
May has said Khan is doing a good job, echoing public sentiment across London.
"Tell him I will remember those statements", Trump said. But he dropped a hint of annoyance when asked about Trump's social media jibes. In that role May was directly responsible for the United Kingdom police service.