Among the suspects is Abu Hamza al-Masri, who U.S. officials charge was involved in kidnapping in Yemen and setting up a terrorist training camp in the United States, the BBC reported.
The BBC and its security correspondent, Frank Gardiner, issued a public apology to Queen Elizabeth II for a "breach of confidence." Gardner revealed on the air Tuesday that the queen had told him she had asked a former home secretary why al-Masri was not behind bars.
"This morning on the 'Today' program our correspondent, Frank Gardner, revealed details of a private conversation which took place some years ago with the queen," the BBC said in a statement.
"The conversation should have remained private and the BBC and Frank deeply regret this breach of confidence. It was wholly inappropriate. Frank is extremely sorry for the embarrassment caused and has apologized to the palace."
Al-Masri's extradition will happen "as quickly as possible," Home Office officials said after a European Court of Human Rights declined Monday to refer their cases to the court's Grand Chamber, the last possible course of appeal to prevent their extradition.
Al-Masri has been fighting extradition to the United States for eight years, arguing he would face inhumane and degrading treatment if he was locked up for life without the possibility of parole.
Babar Ahmad, Syed Talha Ahsan, Adel Abdul Bary and Khaled al-Fawwaz are scheduled to be sent to the United States along with al-Masri. Ahmad and Ahsan are alleged to have run a jihadist website in London, while Bary and al-Fawwaz are alleged to have been aides to al-Qaida leader Osama bin Laden in London.
Bary and al-Fawwaz battled extradition by saying they faced inhumane solitary confinement in a special "supermax" prison.
Some in England have campaigned to have the four men tried in Britain because their crimes occurred in Britain.