Mr Campbell was jailed for life in 1991 over the murder of Mr Hoondle, who was shot dead during a botched robbery in Hackney carried out by two men.
His lawyers and supporters say his confession at the time mirrored what he had watched on BBC's Crimewatch.
Mr Campbell was released on a life licence in 2002 and lives in Felixstowe, Suffolk, with the assistance of a close group of friends and helpers.
His supporters added: "We are all in tears. Ollie's life starts now."
Mr Campbell suffered a brain injury when he was a baby. Experts say the injury has badly affected his memory and general ability to process more than basic information, leaving him suggestible and vulnerable to potential manipulation.
His barrister, Michael Birnbaum KC, told the court earlier this year that there were "ample" grounds to find the conviction unsafe - suggesting he was "badgered and bullied" by police into giving a false confession over his involvement.
Mr Birnbaum said that his client's learning disabilities meant he made admissions which were "simply absurd" and "nonsense", and contained a "litany of inconsistencies" against the facts of the case.
Lord Justice Holroyde said in the ruling: "We accept that, considered in the light of the fresh evidence, the rulings might be different."
He continued: "A jury knowing of the fresh evidence would be considering the reliability of those confessions in a materially different context.
"In those circumstances, we cannot say that the fresh evidence could not reasonably have affected the decision of the jury to convict."
The judge said that the decision would be a "heavy blow" to the family of Mr Hoondle, but he trusted "they will understand that we must reach our decisions in accordance with the law, uninfluenced by emotion".
Officers will continue to support Mr Hoodle's family, a Metropolitan Police spokesperson added.
"The murder of Baldev Singh Hoondle in July 1990 was fully investigated by detectives at the time with a range of evidence brought before a jury who convicted the defendant the following year," they also said.
After Mr Campbell was arrested in late 1990, he was interviewed 14 times by detectives as they sought to build a case to prove that the then 19-year-old had pulled the trigger.
Officers at Hackney Police Station recognised that Mr Campbell had learning difficulties and, for the first 10 interviews, local social services had provided an adult to assist him.
His lawyers believe the purported confession may have therefore been inspired by what he had remembered seeing on the TV, as he tried to end the repeated questioning.
Speaking to BBC Newsnight in 2021, Mr Campbell said the trauma of his conviction had destroyed his life, and that he had lost contact with most of his friends and family.
"It’s disgusting what's happened to me," he told the programme. "That's all I've got to say, it's disgusting how different people out there got away with what they've done to me in my life in 1991."
Mr Campbell added: "I'm an innocent person in the justice system and in the prison system. The justice system put me there."
The quashing of the conviction comes 20 years after the Criminal Cases Review Commission (CCRC), the miscarriages of justice body, rejected his pleas for help - and after two inquiries were triggered into how it had handled the case of Andy Malkinson, who was cleared of rape after a 20-year campaign.
A spokesperson for the CCRC said it referred Mr Campbell's case in 2022, having declined to do so in 2005, "in light of new evidence that his vulnerabilities had not been properly understood or explained at trial".
The commission's review involved speaking again to an expert used at the original trial, as well as a second expert who explained "how a modern approach to assessing Mr Campbell would take a more holistic view that considered his background and experience".
"The Court of Appeal judgment makes clear that its decision is based on fresh evidence from recent research work, and that the understanding of the factors which may contribute to a false confession has increased," the CCRC said.
"It also states that there have been important developments in the law relating to admissibility of evidence and in matters of practice and procedure relevant to a fair trial."