The capital was the last city in government hands, but Taliban fighters reached the outskirts on Sunday morning.
Afghanistan is now on the brink of a total Taliban takeover, with president Ashraf Ghani fleeing the country.
The Prime Minister has now recalled Parliament, with MPs set to discuss the crisis from 9:30am to 2:30pm on Wednesday as the situation worsens.
Mr Johnson has insisted the government’s priority is to get British nationals out of Afghanistan “as fast as we can”.
He said: “Our priority is to make sure that we deliver on our obligations to British nationals in Afghanistan, all those who helped the British effort in Afghanistan over 20 years.
“There’s clearly a change of regime now happening in Afghanistan that has implications for the UK presence, for the platform that we’ve had there for some years.
"We’re working very fast on all the UK nationals, on all the consular cases, and they’re coming forward in numbers at the moment.
“It’s just a question of making sure that they’re able to do it over the next few days.”
With Afghanistan on the brink of complete collapse, the lead elements of the British force sent to evacuate the remaining UK nationals were understood to be in the city amid fears it could fall within days or hours.
The Prime Minister also pledged Britain would work with allies to try to prevent Afghanistan again becoming a “breeding ground for terrorism”.
He continued: “I think it is very important that the West should work collectively to get over to that new government – be it by the Taliban or anybody else – that nobody wants Afghanistan once again to be a breeding ground for terror and we don’t think it is in the interests of the people of Afghanistan that it should lapse back into that pre-2001 status.
“What the UK will be doing is working with our partners in the UN Security, in Nato, to get that message over. We don’t want anybody to bilaterally recognise the Taliban.
“We want a united position among all the like-minded, as far as we can get one, so that we do whatever we can to prevent Afghanistan lapsing back into a breeding ground for terror.”
The Prime Minister’s comments came after he called a meeting of the government’s Cobra emergencies committee on Sunday afternoon involving ministers and senior officials to discuss the worsening situation.
It follows the decision of the Biden administration to withdraw the remaining US troops, triggering a collapse of the Afghanistan government.
Labour leader Sir Keir Starmer earlier warned the situation was “worsening by the hour”.
He said: “The immediate priority now must be to get all British personnel and support staff safely out of Kabul.
“The government has been silent while Afghanistan collapses, which let’s be clear will have ramifications for us here in the UK.
“We need Parliament recalled so the government can update MPs on how it plans to work with allies to avoid a humanitarian crisis and a return to the days of Afghanistan being a base for extremists whose purpose will be to threaten our interests, values and national security.”
The Foreign Office had advised more than 4,000 British citizens thought to be in Afghanistan to leave.
The Home Office said in a statement posted on Twitter on Sunday that it had already relocated over 3,300 Afghan staff and their families, adding : “We will continue to fulfil our international obligations and moral commitments.”
SNP Westminster leader Ian Blackford MP warned the consequences would be felt “well beyond the borders of the country”.
He said: “The UK – along with our international partners – have a moral duty to the people and government of Afghanistan. There are serious questions over the manner of departure from the country, lack of support for the Afghan government, and the reckless cut to aid support.
“Make no mistake about this – the absence of a proper strategy and meaningful planning makes this a serious failure of leadership and one of the biggest foreign policy disasters of modern times.
“I welcome the recall of Parliament to hold the UK Government to account, to discuss next steps to prevent an all-out humanitarian crisis, and to ensure the UK discharges its obligations for all those Afghan citizens who have worked with UK forces.”
Commons foreign affairs committee chair Tom Tugendhat labelled it “the biggest single foreign policy disaster” since Suez.
The Tory MP explained: "We haven't heard from the foreign secretary in about a week, despite this being the biggest single foreign policy disaster since Suez, so I don't know what the Foreign Office is thinking.
“This isn’t just about interpreters or guards. This is about those people who we trained in special forces to serve alongside us, those who helped us to understand the territory through our agencies and our diplomats.
“This is the people who, on our encouragement, set up schools for girls. These people are all at risk now.
“The real danger is that we are going to see every female MP murdered, we are going to see ministers strung up on street lamps.”
Defence committee chairman Tobias Ellwood said it was a humiliation for the West.
He explained: "We assembled the most incredible, technologically advanced alliance the world has ever seen and we are being defeated by an insurgency that’s armed with AK47s and RPGs.
“This will be the biggest own goal made by the West so far this century.”
“My problem is that all the good that we have done is likely to be undone by this dreadful organisation, the Taliban, who will almost certainly impose their very extreme views.
"And the people I feel sorry for are all those women that we introduced to education. Young girls; showing them there is a better way and a way to live your lives that may now in turn have that taken away from them.”
The lead elements of a 600-strong UK force, including Paras from 16 Air Assault Brigade, were understood to be in Kabul to assist with the operation.
The Foreign, Commonwealth and Development Office said the ambassador, Sir Laurie Bristow, remained in the Afghan capital, although the UK diplomatic presence had been reduced.
Earlier Liberal Democrat Leader Ed Davey had written to the Prime Minister calling on him to host a "Westminster leaders’ crisis meeting" on the Afghanistan crisis.
In the letter, he said: "It is without doubt that we face a crucial point in history and, as a nation, we must act together before it is too late.
"The UK has a responsibility to the people of Afghanistan and to the international community.
"Now is the time to act – to do the right thing, and bring political parties together in our national interest."