Britain plans to complete its evacuations out of Afghanistan "in a matter of hours" and no more people will be brought forward, Defence Secretary Ben Wallace said.
Forces have entered the final stage of the evacuation, he said.
He declined to give an exact timeline for the exit of British forces but acknowledged it would come before the Americans withdraw, with US President Joe Biden having set a departure date for Tuesday, August 31.
He said the threat from terror groups such as ISIS-K will only “grow the closer we get to leaving”.
"We will process those people that we have brought with us, the 1,000 people approximately inside the airfield now," Mr Wallace told Sky News.
"And we will seek a way to continue to find a few people in the crowd, where we can, but overall the main processing has now closed and we have a matter of hours."
Mr Wallace said the UK had "closed the Baron Hotel" at 0330 GMT, where those wanting to fly to Britain had gathered.
It also closed the processing centre for asylum seekers and shut Abbey Gate, one of the points of access to Kabul airport, and one of the areas targeted in the twin suicide bombings on Thursday.
Nearly 14,000 British citizens and Afghans had been rescued as part of the UK evacuations since mid-August, Wallace said, but added: "The sad fact is not every single one will get out".
Twin suicide bombs ripped through crowds outside Kabul airport on Thursday, killing scores of people, including 13 US troops and deepening panic in the final days of an already frenzied evacuation effort from Taliban-controlled Afghanistan.
The bombings, claimed by the Islamic State group, left scenes of carnage outside the airport where thousands of Afghans desperate to flee their country had massed.
Mr Wallace said the "horrendous" attacks "didn't hasten our departure".
"We closed the Baron Hotel almost exactly on schedule. The threat is obviously going to grow the closer we get to leaving."
Hours after the attacks, Prime Minister Boris Johnson said Britain would continue to fly its citizens and eligible Afghans from Kabul despite the "barbaric" bomb attack.
"We've been ready for it," Mr Johnson said of the attack. "We are going to continue with that operation, we are now coming towards the very end of it in any event.
"We are going to work flat out ... getting people through as fast as they can still, and we are going to keep going up until the last moment."
Earlier, he had chaired a meeting of the national security emergency committee.
The threat of a terrorist attack was "one of the constraints that we've been operating under" during the operation, he said.
"But clearly, what this attack shows is the importance of continuing that work in as fast and as efficient manner as possible in the hours that remain to us, and that's what we're going to do."
About 15,000 Britons and Afghans who assisted the country during the war have been flown out, said Mr Johnson.
One explosion hit the Baron Hotel, about 200 metres from the Abbey Gate, which had been used by some western nations as a staging point for relocations since the evacuations began on August 14.
After the attack, the British government issued a "Notice to Aviation", advising airlines to avoid Afghan airspace under 7,600 metres.
UK Foreign Secretary Dominic Raab spoke to US Secretary of State Antony Blinken to express Britain's sorrow that US troops lost their lives in Kabul, and that more have been injured.
Mr Raab said they had "paid the ultimate sacrifice while helping others reach safety".
“I also want to extend my condolences to the families of all those Afghans killed or injured. It is tragic that as they sought safety they have suffered at the hands of terrorists," he said.
“Today’s attack is a stark reminder of the dangerous situation in which UK military and civilian personnel have been working so hard to evacuate people, and we should be proud of their bravery and what they have achieved.
"The UK and US remain resolute in our mission to get as many people out as possible. It is testament to the remarkable courage of our personnel that they continue to do so while under fire.
"We will not let the cowardly acts of terrorists stop us.”
