Haroon Aswat is wanted for allegedly conspiring with Hamza, convicted in the US earlier this year on 11 kidnapping and terrorism charges, to establish a training camp in Oregon.
Aswat suffers from paranoid schizophrenia and was transferred to a psychiatric hospital in 2008, the High Court in London heard.
But Lord John Thomas, the head of the judiciary, and fellow senior judge John Mitting said they were satisfied with assurances from US authorities about how he would be treated in custody.
Last year, the European Court of Human Rights (ECHR) held that Aswat's extradition would breach an article prohibiting inhuman and degrading treatment.
Hamza, the former firebrand imam of Finsbury Park mosque in north London, was extradited from Britain in 2012 following an eight-year process delayed by appeals to the ECHR.
He was convicted by a New York court in May on 11 kidnapping and terrorism charges.
The charges related to the 1998 abduction of 16 Western tourists in Yemen, providing material support to Al-Qaeda, assisting the Taliban and sending terror recruits to Afghanistan.
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