Osborne, a 47-year-old father-of-four from Cardiff, was arrested after pedestrians were targeted by a man driving a van near Finsbury Park Mosque in north London early on Monday.
In a statement on behalf of his, Osborne's nephew Ellis Osborne, 26, 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."
Nicola Osborne added: "I'm sorry that my brother has been that troubled that it has taken him to this level of troubledness.
"He has just been troubled for a long time."
Ellis added that his uncle was "not a racist" and said he had never expressed any racist views.
He added: "It's madness. It is obviously sheer madness."
The London Ambulance Service took nine people to three London hospitals. Two others were treated for minor injuries at the scene.
Witnesses described hearing the driver, who was detained by members of the public at the scene, shout: "I'm going to kill Muslims."
He was initially arrested on suspicion of attempted murder but Scotland Yard said he was later arrested for the commission, preparation or instigation of terrorism including murder and attempted murder.
Residents near Cardiff said they were "shocked" after seeing photographs of their neighbour being arrested in London.
In a telephone interview with ITV News, his mother described him as a "complex" person but said "that's all I can say".
She said: "It's a terrible, terrible shock."
Police have been searching a residential address in Pentwyn, where Osborne is listed as living.
Five residents identified images of the man being arrested as their neighbour, Osborne.
Neighbour Khadijeh Sherizi said: "I saw him on the news and I thought 'oh my God' that is my neighbour.
"He has been so normal. He was in his kitchen yesterday afternoon singing with his kids.
"He was the dad of the family. He has kids. He lives next door. He seemed polite and pleasant to me."
The attack unfolded while a man who had taken ill was receiving first aid from the public. The van was driven at people who were helping him.
He was pronounced dead at the scene at 1:04am.
Imam Mohammed Mahmoud was hailed for his efforts to prevent a mob attack and calm the situation before police arrived in shielding the suspected terrorist from the fury of onlookers.
He said: "By God's grace we managed to surround him and protect him from any harm.
"We stopped all forms of attack and abuse towards him that were coming from every angle."
Witnesses said the suspect was smiling and waving as he brought carnage to Seven Sisters Road, with video posted online showing him give a nonchalant wave as police put him in the back of their vehicle.
Osborne, who is believed to have been from Weston-super-Mare, remains in custody at a south London police station.
The attacker, who is believed to have acted alone, struck as the area was busy with worshippers attending Ramadan night prayers at the mosque.
Nine people were taken to hospital, with one since discharged, while two others were treated at the scene.
Metropolitan Police Commissioner Cressida Dick said the incident was "quite clearly an attack on Muslims" and that "we treat this as a terrorist attack".
Flowers have been laid near the scene, with one card reading: ''This is an attack on all Londoners and on my community.''
Another read: ''I was so sorry to hear the news this morning. #NotInMyName''.
Scotland Yard said detectives currently believe the attacker "acted alone but we are of course investigating all the circumstances leading up to the attack".