The Scotsman newspaper yesterday reported the response had been delayed more than four hours as officials debated who would be in charge.
It claimed Kabul police chief General Mohammad Ayub Salangi refused to let commandos take over the mission at 5.40am (local time).
Afghanistan's elite Crisis Response Unit (CRU) and their New Zealand SAS force mentors were called back from a barracks on the other side of the city only about 10 am, the paper said.
It took the commandos five hours to clear the building and kill the insurgents.
During the operation, an insurgent's bullet hit a gap in Corporal Grant's body armour under his armpit and passed through his heart.
Although he clung to life for some time, he died of his injuries as he was being evacuated to Bagram Military Hospital.
The director of Afghanistan's Special Forces, General Abdul Ghafar Sayedzada, reportedly said his squad of commandos arrived at the British Council Building about 20 minutes after a first explosion.
He was turned back after a conversation with the police chief.
"The situation got worse because of the police chief. All the most dangerous positions were occupied by the terrorists," he said.
The paper also revealed that New Zealand SAS forces had been used to blast a way into the compound amid intense machinegun fire.
Afghan and New Zealand forces then "attacked from all sides" until they could rescue two female teachers and a bodyguard who were sheltering in a panic room.
New Zealand's Chief of Defence Force, Lieutenant-General Rhys Jones, said he had no information regarding the claims but was seeking "confirmation and a report from theatre".