In a statement Mr Osborne's nephew Ellis, 26, said "we are massively shocked; it's unbelievable, it still hasn't really sunk in".
A van has struck a crowd outside Finsbury Mosque Park in north London, resulting in one dead and eight injured, AFP reports.
His Muslim neighbour, Khadijeh Sherizi, said she had never had any problems with Osborne until this weekend, when he insulted her children.
After being seized, he said he had wanted to kill "many Muslim people", one witness told journalists.
He was arrested on suspicion of the commission, preparation or instigation of terrorism including murder and attempted murder. The police searched his house in Cardiff yesterday.
Ms Sharazi said she believed the comment was not meant with ill intent and was the first conflict they had encountered with their neighbour.
He said the response of the community had been "extraordinary" starting from the people who were attending the man who died and those who prevented anyone injuring a suspect so that "he was properly arrested, taken away". Hours after the London attack, a man in Paris drove his auto into a police auto; only the attacker died, but it underscored the difficulty of defending against violence by vehicle.
British Prime Minister Theresa May met with members of the Muslim community even as they denounced a rising climate of anti-Islamic sentiment.
Mamhoud was joined by several other bystanders who formed a protective ring around the suspect until police hustled him into the back of a van, the Evening Standard reported.More news: Tillerson urges Qatar and boycotting Arab countries to resolve crisis
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He is being held at a south London police station after being apprehended at the scene by witnesses who claimed to have heard a man shout "I'm going to kill all Muslims" during the attack.
Former friends of the alleged attacker said he grew up in Weston-super-Mare before moving to Wales with his partner and children.
"I saw him on the news and I thought "Oh my God" that is my neighbour".
"I'm not going to defend him, but it's my son, and it's a bad, awful shock and it's not just robbing the bank. Tomorrow when I come to the mosque I'll be looking over my shoulder because two guys are still out there", he added.
Metropolitan Police said they were dealing with a "major incident".
Wellwishers of different backgrounds have said "I stand with you, nobody can destroy such a strong community" and that "love will always be stronger than hate".
Scotland Yard said extra police patrols would be deployed to reassure the public and those observing Ramadan.
British Security Minister Ben Wallace said authorities were aware of rising far-right activity, but were not aware of the suspect prior to the attack near a north London mosque.