On Monday, the court ruled against the defendants' appeal that they would face inhumane treatment if transferred to the United States.
The British government welcomed the ruling and said it would work to ensure the extraditions would happen “as quickly as possible.”
The Egyptian-born Hamza has been serving a seven-year sentence in Britain for inciting his followers to attack non-Muslims.
He is wanted in the United States on a group of charges, including establishing an al-Qaida-style training camp in the northwestern state of Oregon. A federal grand jury also charged him with providing support to terrorists, taking hostages in Yemen and conspiracy to supply goods and services to the Taliban.
Hamza is the former imam of the Finsbury Park mosque which he transformed in the 1990s into the heart of Islamic radicalization in Britain. VoA.