The Libyan refugee briefly came to the attention of MI5 last year and was released from prison earlier this month after he was convicted of non-terror offences.
A mental health alert was raised when Saadallah was not home on Friday evening, and a “street triage” team of police officers and psychiatric nurses later found him in a street in Reading, the Daily Mail reported.
The team reportedly took him home to his council flat before midnight on Friday, just hours before he is alleged to have carried out the attack.
The victims’ families paid tribute in a statement released by police on Tuesday.
Mr Wails’ parents said: “David was a kind and much-loved son, brother and uncle who never hurt anyone in his life. We are broken-hearted at losing him and in such a terrible way.
“We will treasure our wonderful memories of him and he will always be with us in our hearts.”
Mr Furlong’s parents, Gary and Janet, said: “James was a wonderful man. He was beautiful, intelligent, honest and fun.
“He was the best son, brother, uncle and partner you could wish for. We are thankful for the memories he gave us all.
“We will never forget him and he will live in our hearts forever.”
And Mr Ritchie-Bennett’s father, Robert Ritchie-Bennett, said: “I was absolutely blessed and proud to be Joe’s father for 39 years and we are heartbroken by what has happened.”
A note read: “The Blagrave Arms management and staff are devastated at the announcement that the three people who died in the Forbury Gardens attack on Saturday were regular customers and very dear friends of ours.
“Our hearts go out to their family and friends, and the other victims of this horrific incident.”
Michael Main, a friend of the three victims, described the “amazing, caring and beautiful men” he had known for six years.
Speaking to BBC Breakfast, he recalled Mr Wail’s “dry sense of humour” and how “he would do anything for anyone that he could”.
Mr Main said Mr Furlong was a “passionate” and “inspirational” teacher, while Mr Ritchie-Bennett was a “funny and a happy-go-lucky, true and unique individual”.
Any ideology or motivation behind the attack is still unclear, the PA news agency understands.
Saadallah was tackled to the ground by an unarmed police officer who has only been in the force for 14 months, and arrested on suspicion of murder.
He was later re-arrested under Section 41 of the Terrorism Act, which grants powers to hold him without charge for up to 14 days.
On Monday, detectives were granted a warrant of further detention until Saturday. Police have said they are not looking for anyone else in connection with the attack.
Early on Tuesday, Scotland Yard’s head of counter-terrorism policing called those who assisted victims of Saturday’s attack “heroes” who inspired others to “step forward and play our part”.
Assistant Commissioner Neil Basu also encouraged members of the public to view the Action Counters Terrorism (ACT) website and complete the CT Policing online course.
Three people who were injured and treated in hospital had been discharged by Monday evening.
PA understands from security sources that MI5 had received intelligence that Saadallah planned to travel abroad, possibly for terrorism purposes, but the threat was found to be insubstantial and the information provided did not meet the threshold of investigation.
Meanwhile, light has been cast on the fact that the number of people on an MI5 watchlist has risen by thousands.
A Government document from March this year titled Transparency Report: Disruptive Powers 2018/2019, said MI5 was investigating approximately 3,000 subjects of interest (SOIs) across 600 priority investigations.
It said as soon as MI5 judged an SOI no longer posed a threat, that SOI was downgraded and placed in a “closed” category called Closed Subject of Interest (CSOI).
“This does not mean these SOIs will never pose a threat again, but merely that their current level of threat is not judged to be sufficient to prioritise allocating investigative resource against them,” the report said.
It goes on to say the public figure for the number of CSOIs in 2017 was 20,000, and that there are now currently more than 40,000 CSOIs.
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