Archive footage of the radical Islamist cleric Abu Hamza, who will be extradited to the US after a ruling by European court of human rights Reuters and ITN

The European court of human rights has cleared the way for the extradition to the United States of five terrorism suspects, including Abu Hamza al-Masri and Babar Ahmad, after legal battles dating back to 2004.

The decision was immediately welcomed by the home secretary, Theresa May, who said the Home Office would work to hand over the five to the US authorities as quickly as possible.

The home secretary will be keen to avoid the confusion that delayed the removal of Abu Qatada to Jordan earlier this year. The five suspects are expected to be on a plane within weeks.

The suspects involved include the radical cleric Abu Hamza, 54, who is wanted by the US in connection with plans to establish a terrorist training camp in Bly, Oregon, as well as allegations that he provided material support to the Taliban. He is also wanted in connection with allegations that he was involved in hostage-taking in Yemen in 1998.

Hamza, who has lost one eye and a hand, possibly fighting Soviet troops in Afghanistan, was first arrested in London at the request of the US in 2004. But his extradition was halted after he was jailed for incitement offences relating to his sermons at the Finsbury Park mosque in London.

A panel of five human rights judges sitting in Strasbourg rejected appeals to the court's grand chamber from the five suspects and agreed an earlier ruling that their human rights would not be violated by the prospect of life sentences and solitary confinement in an American "supermax" prison. The ruling was seen as one of the most important since 9/11.

All the suspects said they would face inhumane and degrading treatment if they were extradited to the US.

The other four suspects include Babar Ahmad, aged 37, who was first detained in 2004 and is one of the longest-held suspects in detention in Britain without facing trial.

The other cases involve Ahmad's co-accused, Syed Talha Ahsan, and Adel Abdul Bary and Khaled al-Fawwaz, who are accused of being key aides to Osama bin Laden in London.

Ahmad and Ahsan are accused of being involved in a website that encouraged terrorism and which, while operated from London, was hosted in the US.

A Home Office statement said: "The home secretary welcomes today's decision not to refer the cases of Abu Hamza and four others to the grand chamber. This follows the judgment of the European court of human rights on 10 April to allow the extradition of these five terrorism suspects to the US.

"We will work to ensure that the individuals are handed over to the US authorities as quickly as possible."

But the family of Babar Ahmad called for him to be prosecuted in Britain. They said the judges' decision was largely irrelevant as the matter would never have got to this stage had the British police done their job almost nine years ago and provided the material seized from Babar's home to the Crown Prosecution Service, rather than secretly passing it to their US counterparts.

The family's statement said: "The CPS is now in possession of all that material which forms the basis of the US indictment and should immediately prosecute Babar for conduct allegedly committed in the UK.

There is enormous public interest in Babar being prosecuted in the UK, as reflected by the fact that almost 150,000 members of the British public signed a government e-petition to this effect last year. A British businessman, Karl Watkin, has recently commenced his own private prosecution of Babar based on the principle of the matter."

Watkin, a Newcastle-based businessman, said earlier this month that he wanted to bring a private prosecution against Ahmad and Ahsan, rather than outsource the UK's criminal justice system to the US.

The judges said that between 1999 and 2006 the five men were indicted on various terrorism charges in America. Hamza 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.

Bary and 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. Fawwaz faces more than 269 charges of murder.

Ahmad and Ahsan are accused of offences including providing support to terrorists and conspiracy to kill, kidnap, maim or injure persons or damage property in a foreign country.

In April the judges said: "Having regard to the seriousness of the offences in question, the court did not consider that these sentences were grossly disproportionate or amounted to inhuman or degrading treatment." They said "supermax" jail inmates, albeit confined to cells for the "vast majority" of their time, were provided with services and activities – television, radio, newspapers, books, hobby and craft items, telephone calls, social visits, correspondence with families, group prayer – which went "beyond what was provided in most prisons in Europe".

Hamza arrived in Britain from Egypt 28 years ago and worked as a bouncer in a Soho nightclub. Born in Alexandria, he studied civil engineering and in 1984 married a British woman, Valerie Fleming. But in the 1980s he began to turn towards a fundamentalist interpretation of the Qur'an. In 1990 he divorced his wife and returned to Egypt, where he reinvented himself as a Muslim sheikh, or preacher. He travelled to Pakistan and then on to Afghanistan which was at the time gripped by civil war as differing factions fought to fill the power vacuum left by the retreat of Russian troops.

It is unclear if he fought there but when he returned to the UK with his British passport in the early 1990s he was missing his right hand and an eye. In 1996 he re-emerged at Finsbury Park Mosque in north London preaching jihad to a young congregation.

Following the September 11 attacks, he said: "Many people will be happy, jumping up and down at this moment." He was jailed in February 2006 for seven years for soliciting murder and inciting racial hatred and is being held in the maximum-security Belmarsh jail in Woolwich, south-east London.