But the men had achieved their apparent goal of turning a balmy summer night in Britain's capital into a scene of carnage.
Attacking people after abandoning the van, they headed to Borough Market where the pubs and restaurants were packed with Saturday night crowds, many watching the Champions League final between Juventus and Real Madrid in Cardiff.
"I want to express my deepest gratitude on behalf of all Londoners to the police and emergency services for the speed and bravery of their response". The wounds are not considered life-threatening.
The death toll of seven did not include the three attackers. Grande and other stars performed Sunday night at a benefit concert for victims under tight security in Manchester.
There was no immediate claim of responsibility for the London Bridge attack.
That sentiment was echoed by opposition Labour Party leader Jeremy Corbyn, who said "those who wish to harm our people, divide our communities and attack our democracy will not succeed".
May said that Britain was facing a new threat from copycat attacks, and "it is time to say 'enough is enough.'" She called for worldwide agreements to curb online extremism, said longer jail sentences might be needed for terrorism offenses and argued that there is "far too much tolerance of extremism in our country".
She said police believe there were three attackers at London Bridge and Borough Market and that all had been "neutralised".
Corbyn said Britain's democratic values must be maintained.
The sequence of attacks has at times overshadowed the election.
The first victim of the London terror attack has been named as Canadian national Christine Archibald. It had been raised to critical after the Manchester attack, then lowered again days later.
The assault came days before national elections this week and after the British government had downgraded the threat level to "severe" from "critical", meaning that an attack was highly likely, but not imminent.
"We don't believe there are additional elements" at large, said Home Secretary Amber Rudd.
The names and nationalities of those who were killed have not been released.
Greater Manchester Police said they had now arrested an 18th person, a 20-year-old man, in relation to the Manchester bombing.
"So in the light of what we are learning about the changing threat, we need to review Britain's counter-terrorism strategy to make sure the police and security services have all the powers they need".
21of those injured are in a critical condition, the NHS, England's health authority, said. Among the wounded were German, French, Spanish and Australian citizens, officials said.
Assistant Commissioner Mark Rowley said a member of the public also received a gunshot wound in the hail of bullets from the eight officers. "The Uber stopped, we asked people what was going on - people said there was shooting", said Yoann Belmere, 40, a French banker living in London.
Large areas around London Bridge and Borough Market remain cordoned off.
Renan Marquese, a sous-chef at a tapas restaurant, said he was working when he heard chaotic sounds outside the restaurant.
"When I open the door I see three dead people on the floor", he said. "People running everywhere, police shouting to run away".
"There was a vehicle with a loudspeaker saying "go, go" and they (police) threw a grenade. and then I ran", he said.
"The longer you wait and deliberate. the chances of your survivability really diminish notably", he said.
The men drove a van into pedestrians on the city's famous London Bridge. Police cars screamed past the attackers toward the van they had abandoned.
"I saw a young girl being stabbed in the chest", he told LBC radio, recalling how the three "went randomly along Borough High Street stabbing people".