The decision in Strasbourg was welcomed by Prime Minister David Cameron, who admitted to being frustrated by how long the process had taken.
Hamza, 53, was already serving a seven-year sentence for soliciting to murder and inciting racial hatred, while one of his co-accused Babar Ahmad, 37, an alleged terrorist fundraiser, has been held without trial for almost eight years.
They, along with Syed Ahsan, Adel Abdul Bary and Khaled al Fawwaz, will now be extradited to the US to face trial in three months, unless they can secure an appeal in the grand chamber of the European Court of Human Rights.
The court asked for more information about a sixth man, Haroon Rashid Aswat, “on the relevance of his schizophrenia and detention at Broadmoor Hospital to his complaint concerning detention at ADX [Administrative Maximum Facility prison]”.
Aswat is allegedly an accomplice of Hamza, while Ahsan is accused of helping Ahmad set up the pro-jihadist website Azzam.com. Bary and al Fawwaz were allegedly key aides to Osama bin Laden in London.
The men tried to claim that extradition would violate Article 3 of their human rights.
In particular, they argued that being jailed in a “supermax” prison, where they would be held almost entirely in solitary confinement and locked up for 23 hours a day, would be “inhuman and degrading treatment”.
But the judges in the European Court said: “Having fully considered all the evidence from both parties, including specifically prepared statements by officials at ADX Florence, as well as letters provided by the US Department of Justice, the court held that conditions at ADX would not amount to ill treatment.”
The Home Secretary said the UK government “will work to ensure that the suspects are handed over to the US authorities as quickly as possible”.
She added: “I welcome the decision of the European Court of Human Rights to allow the extradition of Abu Hamza and other terror suspects.
“In five of the six cases, the court found that extradition would not breach their human rights and, in the remaining case, it asked for further information before taking a final decision.”
Mr Cameron said he was “very pleased” that the court had rejected the men’s claims that they could face prison conditions and jail terms in the US which would amount to “torture”. Speaking during a trade mission to South East Asia, he said: “I am very pleased with the news.
“It is quite right that we have proper legal processes, although sometimes one can get frustrated with how long they take.”
There have been calls for the procedure to be speeded up.
Keith Vaz, chairman of the Commons Home Affairs Select Committee, said: “Babar Ahmad alone has waited eight years for a decision on his case. This delay is unacceptable.
“We need a system where the Home Secretary is given more discretion over individual cases, ensuring that the important security cases are fast-tracked. I hope she will introduce this when she reviews legislation in the summer.”
However, relatives of the accused called for a public inquiry.
Ahmad’s father Ashfaq said: “British justice appears to have been subcontracted to the US. This should be immediately rectified by putting Babar on trial in the UK… and ordering a full public inquiry into the matter.”
Lawyer Fahad Ansari, who worked on the family’s campaign, said he believed there would be an appeal by the men in the coming months.
He added: “It’s an absolute travesty of justice.”
In a recent TV interview, Ahmad, who admitted to having fought in “battles” in Bosnia, said he believed he was now fighting for his life.
“I am facing extradition to the United States and spending the rest of my life in solitary confinement,” he said.
“I have never been questioned about allegations against me and I have never been shown any evidence against me.
“It is fair to say I’m fighting for my life and I’m running out of time.”
The ruling was criticised by some in Scotland.
Richard Haley, chairman of Scotland Against Criminalising Communities, said: “Today’s ruling is shocking. It’s a slap in the face for Juan Méndez, the UN special rapporteur on torture, who has made it clear that he thinks that prolonged solitary confinement is unacceptable.
“Long-term solitary confinement in any form is cruel and inhuman and should be recognised as a violation of the UN’s Universal Declaration of Human Rights.”
Human rights lawyer Aamer Anwar added: “It is time that our government repealed an extradition treaty that treats our own citizens as second-class citizens and hands them on a plate to the US.”
The European Convention on Human Rights has been controversial in Scotland in recent years, after it played key roles in slopping out compensation payments for prisoners.
It was also instrumental in a change in Scots law, via the Supreme Court’s Cadder judgement, which led to the collapse of almost 900 criminal cases.