Radical cleric can be sent to US to face terrorism charges, along with four other men held in Britain

Abu Hamza, the radical cleric who became the face of violent extremism in Britain, can be extradited to the US to face terrorism charges, the European court of human rights has ruled.

The court in Strasbourg said the human rights of Hamza and four other men held in Britain – Babar Ahmad, Syed Talha Ahsan, Adel Abdel Bary and Khaled al-Fawwaz – would not be violated if sent to the US to stand trial. European judges decided they needed more information about the mental health of Haroon Aswat, an aide to Hamza, before reaching a decision on him.

The ruling clears the last realistic obstacle to Hamza's extradition.

The men have the right to appeal to the grand chamber of the European court and have three months to make an application.

Babar Ahmad's sister said they would appeal the ruling. An appeal has only a slim chance of success, but it could delay extradition for months.

Nazia Ahmad, 27, told the Guardian: "I think we will be taking that avenue." She said the family, from Tooting, south London, would fight on. "We're very disappointed, we're going to carry on fighting for Babar," she said.

Ahmad said her brother should be prosecuted in the UK, and claimed the Crown Prosecution Service had not considered significant parts of the evidence against him, when it decided he should not face charges in Britain.

She said: "His case should have been dealt with by the authorities in the UK. They gathered the evidence, they gave it to the US, and they should have looked at it and decided if he should be put on trial in the UK."

Ahmad said her brother's case represented a "serious abuse of process". She called for a public inquiry and for ministers to answer her famiy's questions.

David Cameron welcomed the court's ruling. 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."

The home secretary, Theresa May, said the government "will work to ensure that the suspects are handed over to the US authorities as quickly as possible".

Hamza, who controlled the Finsbury Park mosque in north London and turned it into a factory for violent jihad, was sentenced to seven years' imprisonment in Britain after being convicted in 2006 of inciting hatred.

The previous government had wanted him sent to the US before his jail term had been completed, but the extradition was halted after his lawyers went to the European court.

The European court of human rights halted extradition proceedings in July 2010, arguing it needed more time to consider complaints that transferring Hamza and others wanted in the US risked breaching their rights by exposing them to possible life imprisonment without parole and solitary confinement.

Ahmad, who the US claims ran a website allegedly raising funds for Islamic extremists, is a computer expert from south London. He has been on remand and refused bail since his arrest in August 2004 on a US extradition warrant.

He was awarded £60,000 in compensation because of the violence British police used during his arrest in December 2003, during which he was punched, kicked and throttled. His case has been supported by the former transport minister Sadiq Khan, who is a family friend and Labour frontbencher.

In the Hamza case, the US has had to given written assurances that it will not impose the death penalty or place the suspects before Guantánamo Bay-style military tribunals.

The US alleges Hamza was in contact with Taliban and al-Qaida terrorists and aided the taking of 16 western tourists as hostages in Yemen in December 1998, an incident that ended in the deaths of three Britons. He is also charged with attempting to set up a training camp for "violent jihad" in Oregon in 1999, along with Aswat.

The court in Strasbourg had considered the cases of the four men for over four years.

In a press release, the court said: "The European court of human rights held, unanimously, that there would be: no violation of article three (prohibition of inhuman and degrading treatment) of the European convention on human rights as a result of conditions of detention at ADX Florence (a "supermax" prison in the US) – if Mr Ahmad, Mr Ahsan, Mr Abu Hamza, Mr Bary and Mr Al-Fawwaz were extradited to the USA; and, no violation of article three of the convention as a result of the length of their possible sentences if Mr Ahmad, Mr Ahsan, Abu Hamza, Mr Bary and Mr Al-Fawwaz were extradited."

The court said it wanted more information on Aswat before reaching a judgment: "The court adjourned its examination of Mr Aswat's application as it required further submissions from the parties, on the relevance of his schizophrenia and detention at Broadmoor hospital to his complaint concerning detention at ADX."