Mr Downey was arrested last year at Gatwick Airport on his way to Greece in connection with the bombing which killed four soldiers and injured 31 others.
But last month the Donegal man walked free from the Old Bailey when the case against him collapsed.
It emerged in court that the 62-year-old was sent a letter wrongly telling him that he was not wanted over any crimes - despite an arrest warrant having being issued by the Metropolitan Police.
Mr Downey, who was convicted of IRA membership in 1974, strenuously denied the allegations against him regarding the bomb attack.
Breaking his silence for the first time, he told the Donegal Democrat newspaper he does not know why he was arrested and that this was a "breach" of agreement between the British and Irish governments.
He said he had travelled back and forth from airports in England and had never been detained before - he had even flown to Canada a year after he received his 'comfort letter'.
"I am told that the reason for my arrest was that I had come up on the Police National Computer (PNC)," he told the paper.
"I had been in and out of Birmingham airport on numerous occasions. I'd also been in Stansted and then on that last day they decided to arrest me. I refuse to believe that if I was on the PNC that I would have gone through all those airports including Derry and Belfast, because that is within their jurisdiction, without being picked up."
The letter states to me that I am not wanted by the PSNI or any police force in the UK.
After the judge threw the case out, a major a political row ensued as it emerged a deal was struck between the last Labour government and Sinn Féin which saw 187 letters issued to republicans, including 38 since 2010.
Prime Minister David Cameron launched a judicial review into the Northern Ireland Office (NIO) scheme after the DUP First Minister Peter Robinson threatened to resign.
NI Secretary of State Theresa Villiers clarified that OTRs who have received letters will not avoid questioning or prosecution if evidence emerges in the future.
While Sinn Féin's Gerry Kelly, who was involved in the process dealing with on-the-runs, has insisted that the letters should not be revoked because those who received them had applied through the system put in place for them.
Mr Downey claimed that unionist representatives knew about the deal.
"As far as the unionists saying that they didn't know about the letters, of course they knew," he said.
"I got the letter in 2007, having applied through Sinn Féin in 2003, four years later the application was granted as part of an ongoing process," he said.
He said that he received his letter a few days after the Northern Ireland assembly was set up.
Mr Downey stressed his support for the peace process and added that his homecoming was never intended to hurt the families of those bereaved - a party planned for his return almost a fortnight ago was called off.
"We need to move forward peacefully together," he said.
"At the end of the day, I am a republican who wants to forward in peace and harmony with the unionist community."
Unionist representatives disputed Mr Downey's claims that they knew about the OTR scheme.
UUP MLA Danny Kinahan, whose friends were killed in the Hyde Park Bomb, said: "My friends and colleagues were murdered on 20 July 1982 and their IRA killers remain at large. Instead of trying to deflect blame in other directions, Downey's time would be better spent telling everything he knows."
DUP MEP Diane Dodds said: "It is absolutely clear that the only people who were aware of this grubby little deal, aside from those in Government who usurped the role of Parliament to institute it, were Sinn Féin and their confidantes."