The leader of Northern Ireland's devolved government Peter Robinson has threatened to quit unless a judicial inquiry is ordered into the collapse of the Hyde Park bomb trial.
Prime Minister David Cameron condemned actions that led to the withdrawal of the prosecution case as a "dreadful mistake".
John Downey, 62, from Donegal denied murdering four soldiers in the 1982 bombing.
The case ended because government officials mistakenly told him in 2007 he was no longer a wanted man in what the victims' families branded a "monumental blunder".
Mr Robinson said he had been left in the dark over secret government letters of assurance given to more than 180 Irish republican paramilitary suspects which led them to believe they would not be prosecuted.
The First Minister said: "I am not prepared to be the person who heads up a government not knowing matters which are totally relevant, completely relevant to the job that we are doing, having responsibility for policing and justice.
"The despicable way that the Government has treated the institutions in Northern Ireland shows that they don't uphold the institutions in Northern Ireland."
Despite regularly travelling to the UK and Northern Ireland, peace process supporter Downey was arrested at Gatwick Airport en route to Greece last May and charged with the murder of four British soldiers and causing an explosion.
He had received a letter of assurance from the Northern Ireland Office (NIO), following Police Service of Northern Ireland (PSNI) advice, in 2007.
Mr Cameron told MPs: "We should be absolutely clear - the man should never have received the letter that he received, Downey.
"It was a dreadful mistake and a mistake that we now need to have a rapid factual review to make sure that this cannot happen again.
"But, whatever happens, we have to stick to the principle that we are a country and a government under the rule of law."
Attorney General Dominic Grieve stood by his decision to charge Downey despite the collapse of his case yesterday due to what victims said was a "monumental" police error.
Defence Minister Anna Soubry, a former barrister, said there was no chance of a judicial review.
The judge, Mr Justice Sweeney, threw the case out after Downey's lawyer successfully argued at the 11th hour that the defendant should not go on trial at the Old Bailey. The Crown announced that it would not appeal against the decision.
At an earlier hearing, Henry Blaxland QC warned of the political ramifications in Northern Ireland of pursuing a trial against Downey in such circumstances, saying that the false assurance he received was "not just negligent, it was downright reckless".
In his judgment, Mr Justice Sweeney said there were "very particular circumstances" of the case. The public interest in prosecution was "very significantly outweighed" by the public interest in ensuring that "executive misconduct does not disrepute", and in "holding officials of the state to promises they have made in the full understanding of what is involved in the bargain".
The legal wrangle raised questions with the PSNI which, the court heard, knew about the UK arrest warrant for Downey but did nothing to correct the error of 2007 in sending the letter.
Mr Robinson said he could not be First Minister in the five-party powersharing administration unless he received answers about the letters, which followed the failure by the Labour government to complete legislation on the position of IRA so-called "on the runs" (OTRs) from justice, and added that he had been left in the dark by the British Government.
Mr Robinson said: "I am not prepared to be the First Minister of a government that has found itself having salient facts relevant to matters that are devolved hidden from them.
"That is not acceptable to me.
"I want to have a full judicial inquiry into who knew what, when they knew it and exactly what they did know at the time.
"I also want to ensure that the letters that have been sent out are rescinded."
Policing and justice powers were devolved from Westminster to Stormont in 2010.
The Democratic Unionist Party (DUP) leader added: "To this present day, no secretary of state - even though I am a privy councillor and they could have spoken to and briefed me on a Privy Council basis - n obody mentioned to me that these letters had been sent out to 187 or more individuals, as it may well turn out to be.
"I think that is astounding, I think it is despicable that the Government should behave in such a way."
But the Defence Minister told BBC Radio 4's The World At One that Mr Robinson could not get a judicial review.
"You can't judicially review the decision," she said. "You can appeal it, the prosecution can appeal it. Dominic Grieve has made it clear the CPS has taken the view that these are not the right circumstances to appeal it.
"So, with great respect to Peter Robinson, who is quite right to be very angry, absolutely right to be angry, but we are in a very, very unpleasant, bad situation and, unfortunately, I cannot see any way back from where we are now."
On July 20 1982, a car bomb left in South Carriage Drive killed four soldiers as they rode through Hyde Park in central London to the Changing of the Guard.
The explosion killed Roy Bright, Dennis Daly, Simon Tipper and Jeffrey Young, and injured other members of the Royal Household Cavalry. Seven horses were also killed as the soldiers travelled from their barracks to Buckingham Palace. Another horse, Sefton, survived terrible injuries and became a national hero.
The investigation into the bombing led police to Downey, through fingerprints on parking tickets and a description given by witnesses of two men carrying out reconnaissance in the area before the attack.
An arrest warrant was issued, but it was decided not to seek Downey's extradition from the Irish Republic in 1989, in part due to the lack of strong evidence against him, the court was told.
Then, in 2007, Downey received assurance that he was not at risk of prosecution as part of the scheme run by the Northern Ireland police.
He was one of 187 "on the runs" to seek clarification from the authorities following the 1998 Good Friday Agreement.
Northern Ireland chief constable Matt Baggott apologised to the families, saying: "I deeply regret these failings, which should not have happened."
He said checks were under way on information processed by the force about other similar cases.
Northern Ireland Secretary Theresa Villiers said police in the province should reflect on "the serious error".
President of the Association of Chief Police Officers Sir Hugh Orde, who was chief constable of Northern Ireland at the time of the error, said: "It is a matter of great personal regret that a crucial oversight was made by a senior officer which resulted in erroneous information being sent to Mr Downey by the Northern Ireland Office and thus prejudicing the current indictment."