Britain was coming to terms with the aftermath of its fourth bloody assault in three months following Monday’s van attack on worshippers leaving the Finsbury Park Mosque in north London.
The string of attacks had “bruised but not broken the heart of this great nation”, Home Secretary Amber Rudd said.
Osborne, 47, a father of four from Cardiff, was arrested.
Police believe the suspect acted alone and searches were being conducted at a residential address in the Welsh capital.
Police are treating the incident as a terror attack and British Prime Minister Theresa May described it as “sickening”, vowing Monday to fight extremism in all its forms.
The attack raised fears of retaliation against Muslims after a series of deadly assaults in Britain by Islamist extremists.
One man who was already receiving first aid at the time died following the assault, while nine people were taken to hospital and two others were treated for minor injuries.
“I’m sorry that my brother has been that troubled that it has taken him to this level of troubledness,” said the suspect’s sister Nicola Osborne.
“He has just been troubled for a long time.”
His mother Christine, 72, said she screamed when she saw her son in television footage.
“My son is no terrorist — he’s just a man with problems,” The Sun newspaper quoted her as saying.
In a statement on behalf of his family, his nephew Ellis Osborne, 26, said: “We are massively shocked.
“Our hearts go out to the people who have been injured.”
His uncle was “not a racist”, he said. “It’s madness. It is obviously sheer madness.”
Suspect saved by imam
Londoners bearing flowers and messages of solidarity gathered late Monday at the scene of the attack, some carrying signs reading “United Against All Terror”.
Another vigil is planned for Tuesday.
The van driver was pinned down by locals before being shielded from retaliatory violence by an imam and detained by police.
The man suspected of driving the van was arrested on suspicion of “the commission, preparation or instigation of terrorism including murder and attempted murder”, police said.
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 holy month of Ramadan draws to a close.
Rudd said Muslims needed to feel safe in Britain and the government was working to tackle all forms of hate crime and extremism.
“Indicative figures suggest that over half of those who experience hate because of their religion are Muslim. Any hate crime is unacceptable but this stark figure is something we will not shy away from,” she wrote in The Guardian newspaper.
“We stand with the Muslim community — you are not alone, we share your pain and we will not let you down.”
‘Does not remember anything’
One victim of Monday’s attack has no memory of what happened, according to a nephew who did not wish to be identified.
“He is bleeding out of his ear, but in general his health was stable,” he said after visiting his uncle Hamza Sharif in hospital.
“He has a fracture in his skull — but they still don’t know why the bleeding from his ear is not stopping yet,” the Somali-born man said.
Sharif “does not remember anything” of the attack and kept asking “what was wrong”.
The three previous terror attacks in recent months were all Islamist-inspired.
Two months later, on May 22, 22 people were killed when a suicide bomber blew himself up outside a pop concert in the northern city of Manchester.
On June 3, eight people were killed in a van and knife attack on London Bridge and nearby Borough Market.