In a statement, the International Committee of the Red Cross said the body was transferred to Beirut and "the British Embassy in Lebanon is expected to rapidly fly the body to London."
While Syrian authorities have said Khan committed suicide in jail, his mother Fatima Khan lashed out at Damascus, blaming her son's jailers for his death.
In an interview with the BBC, she said: "How come they can't differentiate between humanitarian aid worker and a terrorist?"
Khan's bereaved mother added: "He did not treat any terrorists, he treated only women and children, he told me ... look I am a doctor mummy, my profession is to give life."
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Like Khan's family, London and a Syrian monitoring group have held Damascus responsible for the doctor's death.
The Syrian Observatory for Human Rights said Damascus frequently claims detainees who have been tortured to death committed suicide.
Syrian authorities said on Wednesday that Khan was found "hanging" in his cell, where he was being held for "unauthorised activities," and had killed himself.
Previously, Deputy Foreign Minister Faisal Muqdad had said the ICRC was present at a final autopsy in Syria on Khan's body, but the organisation's spokesman Simon Shorno said his organisation "did not take part in any autopsy."
Muqdad said President Bashar al-Assad had decided to grant Khan an amnesty and hand him over to his mother and British MP George Galloway at a news conference in Damascus.
The ICRC also said it "does not visit detainees in Syria despite repeated requests to Syrian authorities to be granted access to places of detention in the country. It therefore never visited Dr. Khan during his detention".
The London Metropolitan Police meanwhile told AFP that it will not be investigating Khan's death, but that it is ready to assist the coroner in his inquiry.