The 67-year-old John faced Omagh Magistrates’ Court on Saturday, October 12
Downey was first detained in the Republic in October 2018 under a European arrest warrant.
He fought extradition but lost his most recent appeal, and handed himself in to Gardaí (Irish police) on Friday.
Downey is wanted by prosecutors in connection with the murders of UDR soldiers Alfred Johnston, 32 and a father of four, and James Eames 33, a father of three.
Both were killed in a bomb attack in Enniskillen in 1972.
Downey, from Creeslough, Co Donegal, is also accused of aiding and abetting an explosion.
A crowd of supporters, including Sinn Fein representatives, gathered outside the court ahead of the hearing.
A Detective Chief Inspector told the court police would be objecting to bail as they viewed Downey as a flight risk.
He pointed to the defendant’s repeated attempts to appeal the extradition order.
“The police view would be the defendant is reluctant to face justice,” he said.
In 2013, Mr Downey was charged with murdering four Royal Household Cavalry men in a bombing at London’s Hyde Park in 1982.
The case dramatically collapsed after it emerged the defendant had a letter of assurance from the former Prime Minister Tony Blair that he was not actively wanted by the authorities.
In court on Saturday, the Detective Chief Inspector said that following the collapse of the Hyde Park bombing trial in 2014, the families of those killed in the incident had launched civil proceedings against Mr Downey.
He told the court that Downey subsequently transferred assets including land from his name to his wife’s name.A defence solicitor said his client had handed himself in to Gardaí by agreement and had fully complied with previous bail conditions in 2013 and 2014.
He described the land transfers as a “red herring”, and said his client was a man of ill health.
Judge Michael Ranaghan said he envisaged a “long and protracted” legal process ahead.
But said he found the defendant did pose a flight risk and so bail was refused.
The judge told Downey he can appeal his decision to the High Court in Belfast.
He ordering him to appear again in court on Friaday, October 25.
Downey was led away in handcuffs and taken to the remand unit in maghaberry prison.