Radical cleric Abu Hamza has arrived in the United States after he was extradited from Britain along with four other terror suspects.
Hamza, Babar Ahmad, Adel Abdul Bary, Syed Ahsan and Khalid al Fawwaz were flown to New York on board two planes which left a military airbase in Suffolk late on Friday.
The suspects had made last-ditch challenges against their removal from the UK, which were rejected by two High Court judges in London hours earlier.
Sky's Mark Stone said the judicial process in the US would begin almost immediately.
He said of Hamza: "He will be in a courtroom, probably later on today. That will be just a formality. There will then be a pre-trial hearing which will take place within three weeks.
"His trial and the trial of the other four will probably take place within one to three years."
The US Attorney's office in Connecticut confirmed that Ahmad and Ahsan are scheduled to appear before the US District Court in New Haven.
Prime Minister David Cameron said he was "absolutely delighted" Hamza was out of the country.
"Like the rest of the public I'm sick to the back teeth of people who come here, threaten our country, who stay at vast expense to the taxpayer and we can't get rid of them.
"I'm delighted on this occasion we've managed to send this person off to a country where he will face justice."
He said the Government must consider ways of stopping similar cases in the future.
Home Secretary Theresa May told Sky News: "We've been very clear that we've moved quickly to ensure that once we got the final decision it was possible to remove these individuals from the UK."
The decision comes after a legal battle lasting up to 14 years, which has involved appeals in British and European courts.
Mrs May said she was looking at ways of speeding up the extradition process.
She said: "I have already set some work in train to look at comparisons with other countries ... and we will be looking at perhaps to remove some of the stages of the process."
The terror suspects were transported from HMP Long Lartin in Worcestershire to RAF Mildenhall in a police convoy.
Officers from the Metropolitan Police's extradition unit then handed the men to US officials before they were taken aboard the planes which left British soil just before midnight on Friday.
Sir John Thomas, president of the Queen's Bench Division, and Mr Justice Ouseley on Friday rejected an application by 54-year-old Hamza, a former imam at Finsbury Park mosque in north London, to be given time to undergo a brain scan his lawyers said could show he is medically unfit to face trial.
They also threw out challenges by Ahmad, Ahsan, al Fawwaz and Bary, paving the way for Mrs May to give the go-ahead for their immediate extradition.
All five cases returned to the High Court after judges at the European Court of Human Rights (ECHR) refused to intervene and stop the Home Secretary extraditing them.
A US Embassy spokeswoman said: "These extraditions mark the end of a lengthy process of litigation through the UK courts and the ECHR.
"The US government agrees with the ECHR's findings that the conditions of confinement in US prisons - including in maximum security facilities - do not violate European standards."
Between 1999 and 2006, the men were indicted on various terrorism charges in America.
Hamza, who was jailed in the UK for seven years for soliciting to murder and inciting racial hatred in 2006, first faced an extradition request from the Americans in 2004.
He has been charged with 11 counts of criminal conduct related to the taking of 16 hostages in Yemen in 1998, advocating violent jihad in Afghanistan in 2001, and conspiring to establish a jihad training camp in Bly, Oregon, between June 2000 and December 2001.
Ahmad, a computer expert from south London, and Ahsan are accused of offences including using a website to provide support to terrorists and conspiracy to kill, kidnap, maim or injure persons or damage property in a foreign country.
They wanted their removal stopped so they could challenge a decision by the Director of Public Prosecutions not to allow British businessman Karl Watkin, a campaigner against the UK's extradition arrangements with the United States, to bring prosecutions against them in the UK.
Bary and al Fawwaz were indicted - with Osama bin Laden and 20 others - for their alleged involvement in, or support for, the bombing of US embassies in Nairobi and Dar es Salaam in 1998.