Neil Basu, London's police counterterrorism chief, said 28-year-old Usman Khan was attending a program that works to educate prisoners when he launched Friday's attack just yards from the site of a deadly 2017 van and knife rampage.
As Khan is tackled to the ground, a third individual wrestles the knife out of Khan's hands and backs away, with police closing in with their guns out.
The man was shot dead by police after stabbing and killing a man and a woman and injuring another man and two women, police said.
The latest attack came less than two weeks before Britain's general election, and politicians temporarily suspended campaigning.
Boris Johnson broke away from campaigning on Friday for the December 12 election to rush back to Downing Street for a security briefing on the attack. Last year, he plead guilty to participating in an Al-Qaeda inspired plot to blow up the London Stock Exchange.
Dutch police say they have found no evidence linking the attack to terrorism. Khan left school in the United Kingdom with no credentials and is believed to have spent part of his late teens in Pakistan, where he lived with his mother when she took ill.
They said Khan had been known to authorities in Britain, having been convicted in 2012 for "terrorism offenses".
Footage posted online shows Khan being taken to the ground as one man sprays him with a fire extinguisher and another, reportedly a Polish man who worked at the Fishmongers' Hall, lunges towards him with a narwhal tusk believed to have been taken from the wall inside the building.
Professor Stephen Toope, vice chancellor of the University of Cambridge, said the university was in touch with the Metropolitan Police and awaiting further details of the victims.
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"Everyone was just on top of him trying to bundle him to the ground", tour guide Stevie Hurst, who ran from his auto to the place of incident, told BBC radio.
Police said Khan had previously been arrested in 2012 on terror-related charges. As a result, officers are, tonight, carrying out searches at an address in Staffordshire.
Inmates are usually released half-way through the type of determinate sentence he was given, and time spent in custody before trial may have been taken into account.
The British Parole Board in a statement said it had no involvement in the release of Khan "who appears to have been released automatically on license [as required by law] without ever being referred to the Board".
During the attack, he had been wearing an electronic tag used to monitor criminal offenders, according to a report in British newspaper The Times.
Prime Minister Boris Johnson, who is seeking re-election on December 12, said it was important to enforce terrorism-related sentences.
"I express my enduring thanks to the police and emergency services, as well as the courageous individuals who put their own lives at risk to selflessly help and protect others".
The month before, a suicide bomber killed 22 children and adults and wounded 59 at a packed concert hall in the English city of Manchester, as crowds began leaving a concert by United States singer Ariana Grande.
Johnson also told the BBC that his government would review sentencing policies in the wake of the attack. The suspect was shot dead by police and was wearing what is thought to have been a hoax explosive device.