“How on earth could we have let this guy or possibly more through the net — what happened?” he asked in an interview on Sky News.
Some of the missed warnings were especially glaring because they came from the very people the British government had entrusted with identifying extremists.
Usama Hasan, a former Islamic extremist who now works with the police to help de-radicalize others, said he had a physical altercation in a London park less than a year ago with one of the assailants, Khuram Shazad Butt.
Mr. Butt’s brother, Saad, who did paid work for the police on counterextremism issues and was estranged from the assailant, missed signs of how dangerous his brother’s extremism had become.
Other warnings had also been raised about Mr. Butt, 27, who held odd jobs, including at KFC and a six-month stint as a customer service trainee for the London subway system that ended in October. His second child was born weeks before the attack, neighbors said.
In 2015, an F.B.I. informant, Jesse Morton, wrote a report to his handler in the United States, identifying Mr. Butt as a person to watch because of what Mr. Morton described as his rising role in extremist chat rooms run by Al Muhajiroun, an organization banned in Britain because of its sprawling links to terrorism.
“My handler got back to me and said it was ‘excellent work’ and forwarded it to the head office,” said Mr. Morton, a former Qaeda recruiter from New York who served prison time on terrorism charges before recanting and agreeing to work undercover for law enforcement.
Mr. Morton, who recently started Parallel Networks, an organization combating extremism, said it was unclear to him whether his F.B.I. report had been forwarded to British officials. A spokesman for the F.B.I., Andrew C. Ames, said the agency had no comment.
Even excluding the F.B.I. report, plenty of alarms were ringing for the British authorities about Mr. Butt.
Neighbors and friends noticed his behavior, including a mother of three who lived in the same apartment building as Mr. Butt in Barking, a suburb in eastern London.
The mother, Erica Gasparri, was quoted by The Guardian as saying she confronted Mr. Butt two years ago after he tried to convert her son to Islam. When she found him in a local park, she recalled, Mr. Butt said he was ready to do “in the name of Allah what needs to be done, including killing my own mother.”
Ms. Gasparri said she had called a police hotline and passed on photographs she had taken of him, but never got a call back.
At the KFC where he worked in Barking, employees were on edge after a video surfaced of Mr. Butt alongside other Muhajiroun members sparring with the police, who were called to a London park where the men had unfurled an Islamic State flag, said Ishtiaq Ahmed, whose brother worked at the same restaurant.
That video clip was also featured in a Channel 4 television documentary broadcast last year, “The Jihadis Next Door,” about extremists living in Britain.
Meanwhile, in Italy, the authorities allowed the second attacker, Youssef Zaghba, to walk past them last year at an airport security check, even though he was carrying Islamic State propaganda. Mr. Zaghba, 22, an Italian of Moroccan descent, was en route to Syria to fight for the Islamic State when he was stopped in March at the airport in Bologna. He was traveling on a one-way ticket, and the authorities found Islamic State material on one of his electronic devices, said two former European intelligence officials, who spoke on the condition of anonymity because they were not authorized to discuss the matter.
Giuseppe Amato, the chief prosecutor in Bologna, told a different account to Radio 25 in Italy, saying that Mr. Zaghba had been stopped en route to Istanbul because he was carrying nothing more than a knapsack, which raised suspicion.
“He told the security guard, who checked him — and then he corrected himself — that he was going to be a terrorist,” Mr. Amato said.
Mr. Zaghba was arrested and his belongings were confiscated, but after a judge charged with verifying the accusations against him found there were no grounds to hold him, they were returned, “so the contents on his device” were not examined, Mr. Amato said.
Still, Mr. Amato said that Mr. Zaghba had been singled out as a “suspicious person” to the British authorities.
“We did everything we could do,” he said. “But there was no proof he was a terrorist.”
Free to move around in Italy, Mr. Zaghba tapped into that country’s little-known Muhajiroun network, said a retired senior European law enforcement official who was keeping tabs on the investigation. While the dates are unclear, the official said the Italy-based network appeared to have introduced Mr. Zaghba to Mr. Butt.
Al Muhajiroun has been described as perhaps Europe’s most effective jihadist recruitment machine. An estimated one-third of European Muslims who have joined the Islamic State, or ISIS, in Syria in recent years were influenced by the network of groups spawned by Al Muhajiroun and one of its founders, Anjem Choudary, a lawyer turned radical preacher. In Britain alone, at least half of all terrorism cases have publicly documented links to Mr. Choudary, including the 2005 London transit bombings and the killing of a British soldier, Lee Rigby.
“Muhajiroun has significantly been involved in preparing and recruiting for ISIS,” said Rashad Ali, a resident senior fellow of the Institute for Strategic Dialogue, a London-based research group that focuses on countering extremism.
The third attacker, identified as Rachid Redouane, 30, was denied asylum in Britain in 2009, according to news reports, and moved instead to Ireland, where he married, worked as a pastry chef and lived for a time in Dublin. He apparently managed to sneak into Britain through the porous Irish border to join the other attackers.
Of all the warning signs about Mr. Butt, the most detailed account was offered in an interview on Tuesday with Mr. Hasan, the member of the network of former extremists and the head of Islamic studies at the Quilliam foundation.
Last July 6, Mr. Hasan said, he ran into Mr. Butt at a Muslim family fair in East London. He was standing beside a fairground ride that his 9-year-old son had just mounted, when Mr. Butt, wearing a traditional Muslim robe and headdress, assailed him with abuse. “You take money from the government to work against Muslims. You spy on Muslims,” he raged. “You are a murtadd,” an epithet used by the Islamic State.
Then he tried to charge Mr. Hasan.
“He ran straight for me with his face contorted in hatred,” Mr. Hasan recalled.
A fight ensued with multiple people, including Mr. Butt’s wife, dressed in a face-covering veil. Mr. Hasan reported the episode to the police, emailing them photos of Mr. Butt that one of his family members had taken on a cellphone.
He told them Mr. Butt had displayed all the outward signs of a radicalized political Islamist he knew so well from his own past in radical circles and more recent de-radicalization work: the combination of a pious Islamic dress and long unkempt beard with the angry demeanor and the rehearsed lines.
“I told them I was certain these guys were Al Muhajiroun,” Mr. Hasan said. “I said they are a national security threat. They need to be monitored.”
The police constable on duty assured him that his concerns had been passed on to the Metropolitan Police’s counterterrorism unit, SO15, Mr. Hasan recalled.
Six months later, in January, Mr. Hasan said, he received a phone call. The police had identified Mr. Butt, but no charges were brought. “I am sure this man will sooner or later be arrested in some terrorism plot,” he recalled telling students in May at a lecture at Cambridge University.
On Tuesday morning, after having seen photographs of Mr. Butt in the news media, Mr. Hasan emailed the constable who had dealt with his assault case.
“Could you please confirm that my assailant in the above case (CAD 7943 06JUL16) is the same Khuram Butt who was one of the London Bridge terrorists over the weekend?” he wrote.