Boris Johnson has pledged to do whatever it takes to get as many people out of Afghanistan after August 31 as possible.
Two Britons and the child of another Briton died in the terror attack outside Kabul airport, the Foreign Office confirmed on Friday.
The Prime Minister said: "I think what their loss really underlines is the urgency of getting on and concluding Operation Pitting in the way that we are, and also underlines the bravery of our armed services, our troops, everybody else involved."
He admitted he felt "a great sense of regret" about those left behind in Afghanistan.
"Of course, as we come down to the final hours of the operation there will sadly be people who haven't got through, people who might qualify,” he said at the Foreign Office.
"What I would say to them is that we will shift heaven and earth to help them get out, we will do whatever we can in the second phase."
Asked whether the scenes seen in Afghanistan amounted to a national humiliation, he said the circumstances were "extremely difficult and extremely horrible".
He added: "The timing of this is certainly not the one that this country would have chosen, and I think that everybody understands that.”
Mr Johnson repeated his warning to the Taliban that if any new government in Afghanistan wanted to engage with the West, it must allow people to flee the country if they wish.
He said: "There will be people who are eligible, whether they're UK nationals who have chosen not to come forward yet, or people who were interpreters and others who haven't been able to get to come forward to Hamid Karzai International Airport so far.
"And what I say to them, is that we will shift heaven and earth to get you out, and we will use all the leverage that we have with the Taliban to make sure that they understand it."
Foreign Secretary Dominic Raab said: “I was deeply saddened to learn that two British nationals and the child of another British national were killed by yesterday’s terror attack, with two more injured.
“These were innocent people and it is a tragedy that as they sought to bring their loved ones to safety in the UK they were murdered by cowardly terrorists.
“Yesterday’s despicable attack underlines the dangers facing those in Afghanistan and reinforces why we are doing all we can to get people out. We are offering consular support to their families.
“We will not turn our backs on those who look to us in their hour of need, and we will never be cowed by terrorists.”
