James Heappey said the UK was working to a deadline of 31 August, when the US is set to leave - though the prime minister is expected to ask President Joe Biden for more time.
Evacuations are also dependent on Taliban co-operation, Mr Heappey added.
But he added the UK was "taking nothing for granted" with the militants.
Boris Johnson will ask Mr Biden to extend the deadline for withdrawing US troops at an emergency gathering of world leaders from the G7 countries on Tuesday, in order to allow evacuation flights to continue.
Thousands of people are said to be waiting to board flights at Kabul's international airport, just over a week after the Taliban seized the capital.
Mr Heappey told BBC Breakfast that 6,631 people had been evacuated to the UK in the past week, and there would be nine flights over the next 24 hours.
He said around 1,800 "eligible people" or UK passport holders remained in Afghanistan - as well as 2,275 Afghans who can be resettled having worked for the UK government, and a further list of people from "wider Afghan civil society who we would like to get out if we're able".
He said around 1,800 "eligible people" or UK passport holders remained in Afghanistan - as well as 2,275 Afghans who can be resettled having worked for the UK government, and a further list of people from "wider Afghan civil society who we would like to get out if we're able".
He explained that the evacuations would not be possible without the US, which "has effectively taken over the full operation" at the airport.
"If there is no opportunity to extend [the deadline] - either because there's not the international appetite to do so, or perhaps more likely the Taliban are unwilling to allow us to - then we need to continue with our plans to be out by 31 August," he said
"If that is to be the case, every minute counts to get as many people out in the meantime."
"If there is no opportunity to extend [the deadline] - either because there's not the international appetite to do so, or perhaps more likely the Taliban are unwilling to allow us to - then we need to continue with our plans to be out by 31 August," he said
"If that is to be the case, every minute counts to get as many people out in the meantime."
Later, on BBC Radio 4's Today programme, he said "the period of time it would take to get in place a replacement force is not realistic".
He added that the Taliban also "gets a vote" on whether evacuations continue after August, and that "apparently they've indicated that they wouldn't be" supportive of an extension.
The Foreign Office said it had sent five additional members of staff to Kabul to help with the evacuation, bringing the total number of its staff there to 19.
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