Osborne was arrested on suspicion of the commission, preparation or instigation of terrorism including murder and attempted murder.
Police on Tuesday questioned a man suspected of deliberately mowing down Muslim worshippers in North London, as the interior minister said Britain was "bruised but not broken" by a series of terror attacks.
The suspect was named by British media as Darren Osborne, 47, a father-of-four, who lived in Cardiff. Osborne was arrested on suspicion of the commission, preparation or instigation of terrorism including murder and attempted murder.
British media report that some of Osborne's neighbours have described the father of four as "aggressive" and "strange".
Witnesses said Osborne claimed he wanted to "kill all Muslims" after he drove into a crowd leaving a mosque early Monday morning.
The vehicle swerved into the group of worshippers, mainly of North and West African origin, as they left the Muslim Welfare House and nearby Finsbury Park Mosque as they left after prayers, injuring 11.
Vigil for victims of terror attack
Londoners bearing flowers and messages of solidarity gathered Monday at the site where Osborne ploughed the van he was driving into pedestrians.
Among the roughly 100 people at the vigil, some carried signs reading "United Against All Terror".
"One of the things that all these terrorists share is a perverse ideology that wants to fuel division and divide our communities. We're not going to let them," Mayor Sadiq Khan said, speaking after prayers at the Muslim Welfare House on Monday evening.
Family in shock
Osborne's family said they are devastated at the "madness" of the attack.
TRT World's Alican Alyanlar reports.
In a statement given to local media on behalf of his family, his nephew Ellis Osborne said: "We are massively shocked; it's unbelievable, it still hasn't really sunk in. We are devastated for the families, our hearts go out to the people who have been injured. It's madness. It is obviously sheer madness."
Police presence beefed up
Security Minister Ben Wallace said the man was not known to the security services and police said they believed he was acting alone. They were carrying out searches of addresses in Cardiff where the vehicle hire company that the van was rented from was based.
London police chief Cressida Dick said the incident was "quite clearly an attack on Muslims" and promised a stepped-up police presence near mosques as the Islamic month of Ramadan draws to a close.
"Sickening" terrorist attack
Prime minister Theresa May, who was heavily criticised for failing to meet survivors of a devastating fire in a London tower block last week, visited Finsbury Park Mosque where she met local faith leaders.
May condemned the assault as "sickening", saying Britain's determination to fight "terrorism, extremism and hatred... must be the same, whoever is responsible".
The latest incident took place just over two weeks after three militants drove into pedestrians on London Bridge and stabbed people at nearby restaurants and bars, killing eight.
A suicide bombing at a pop concert in Manchester, northern England, in May also killed 22 people, while in March, a man drove a rented car into pedestrians on Westminster Bridge in London and stabbed a policeman to death before being shot dead. Five people were killed in that attack.