It comes amid concern at mercy flights leaving Kabul with many empty seats on board, despite thousands of people gathering at the airport in a desperate attempt to flee the country.
Mr Heappey admitted: “We don’t have it in our gift to say that it will last for another five days, 10 days. It is clearly circumstances that are very dynamic.
“We are simply doing our best to maximise flow, each and every day. As the Defence Secretary has sadly had to say on a number of occasions, the reality is that eventually the air bridge will have to close and quite possibly not everybody will have been got out.
“That’s what keeps us awake at night. That’s what’s motivating to us to work as hard as we are to ensure that those numbers are the absolute minimum.”
Western countries have vowed to take more than 100,000 Afghan refugees between them but Nato said only 18,000 have been flown out since Sunday. One image laid bare the extent of the situation, showing what is thought to be a Norwegian mercy flight carrying the wife of a British ex-Marine but almost nobody else. Paul Farthing, 57, posted on Twitter an image of the plane used by his Norwegian wife Kaisa.
She boarded a military transport aircraft for the flight to Norway after a hazardous drive to the airport at night.
Mr Farthing said in a tweet that the situation was “scandalous.”
He continued: “Thousands wait outside Kabul airport being crushed as they cannot get in. Sadly people will be left behind when this mission is over as we CANNOT get it right.”
Mr Farthing later told Sky News how aircraft are taking off from the airport every hour “regardless of whether they’re full or not”.
Flights bound for Australia, the Netherlands, France and Italy are reported to have been taking off with just a few dozen people on board, while a German plane with room for 150 departed on Tuesday with just seven passengers.
Giant US cargo planes are said to have been departing with an average of just 100 people on board —despite being able to carry more than 600.
The US State Department, meanwhile, was forced to scrap a plan to bill US citizens and allies $2,000 or more for their evacuation flights amid widespread criticism.
The UK Government has stressed its flights have been full and Mr Heappey today said he understood the Taliban were not turning people away from Kabul airport, adding: “Where they have done, I’ve heard it’s more that they are being officious rather than malicious.”
Mr Heappey said if people are called forward for flights, they should “have confidence” the Taliban will let them through to the handling centre.
He added: “People should not despair that the air bridge is their one route out of Afghanistan to the United Kingdom.
“There will be a second phase to this operation which will see people being able to enrol at other British embassies and high commissions in the region and to be able to be resettled that way.”
It comes after footage emerged of desperate Afghans handing their children to soldiers amid the chaos at Hamid Karzai International Airport.
A total of 12 people are said to have been killed in and around the airport since the Taliban took control of the capital on Sunday. The deaths were caused either by stampedes of people trying to get into the airport in the hope of boarding an evacuation flight or by gun shots in the area.
A government spokeswoman said earlier: “Departments across Whitehall have been working intensively at all levels in the last few days and weeks on the situation.
“Thanks to these efforts, we have relocated more than 2,000 Afghans to the UK since June, evacuated more than 400 British nationals and their families on RAF flights since Sunday and established one of the most generous asylum schemes in British history.”