UK forces will continue with the final stages of an evacuation programme from Kabul despite a deadly suicide bombing and gun attack outside the city’s airport, Boris Johnson has said.

But while the prime minister said the UK had extracted the “overwhelming majority” of UK nationals or Afghans eligible to be removed, a series of MPs warned they knew of large numbers who faced being stranded.

Several MPs said that of hundreds of constituents’ family members seeking to leave, many of them UK passport holders, almost none had made it, and that relatives in the UK were being advised to email advice services that never replied.

Johnson chaired a meeting of the government’s emergency Cobra committee after two powerful suicide bombs and a gunman struck one of the main entrances to the Afghan capital’s international airport.

“We’re going to keep going up until the last moment,” the prime minister said. “The conclusion is that we’re able to continue with the programme in the way that we’ve been running it, according to the timetable that we’ve got.”

He added: “We’re now coming towards the very end of it in any event, and we’ve already extracted the overwhelming majority of those under both the schemes – the eligible persons, UK nationals, the Afghan interpreters and others. And it’s been totally phenomenal effort by the UK. There’s been nothing like it for decades and decades.”

Johnson said that while some people eligible to reach the UK would not be reached before the airlift ended, the UK would pressure the Taliban to let them out – something Afghanistan’s new rulers have said they will not do.

Government figures released on Thursday night said that more than 13,140 people had been evacuated by the UK military since the mission began on 13 August. This included embassy staff, British nationals, those eligible under the Afghan Relocation and Assistance Policy (Arap) programme and some nationals from partner nations.

However, a number of MPs told the Guardian they knew of large numbers of people still in Afghanistan who were eligible to leave for the UK who now had no idea how this would happen.

The Labour MP Lyn Brown, who represents West Ham in east London, said she was “infuriated” with the response she received from the government, calling it “a complete dereliction of duty”.

Over the course of the crisis her office has taken up cases on behalf of 82 people living in her constituency. But most of them have been asking for help on behalf of multiple family members, and so her office has been trying to help 356 individuals reach Britain.

One of the most harrowing cases involved a British girl of school age needing a passport to leave. Brown’s office first raised this with the Home Office early last week, but it took days to get a response and the girl was only told to go to the Baron hotel – where people have been processed ahead of evacuation by the British – as the airlift mission was coming close to the end.

On Thursday afternoon Brown and her staff were told the girl had been on a bus cleared to head for the airport – but that she and others without a passport had then been ordered off by a solider, and left stranded.

Harriet Harman, the Labour MP for Camberwell and Peckham in south London, said that of 228 family members of 34 constituents, including UK nationals, joint UK-Afghan nationals, and relatives of UK nationals, none had managed to leave since the Taliban took over.

While ministers have been briefing MPs, Harman said, it appeared impossible to get news on how people could leave.

“They keep saying, ‘Here’s an email address to get updates,’” she said. “But we have had no feedback aside from the changing kaleidoscope of general advice. We’ve had no feedback on our individual cases. None of us have.

“Really it would be better for the ministers to just say, ‘Sorry, we can’t give you any updates, because we haven’t got the capacity.’ Obviously, there’s a sense of mounting desperation.”

Alison Thewliss, the Scottish National party MP for Glasgow Central, said up to 50 constituents were similarly seeking news. “We’ve had one family who had to move from house to house to house, and then to the airport to wait. That’s women on their own, with five children, one of them eight months pregnant.

“We’re providing the information that we have, but it’s not really specific enough to help people. It’s really difficult. It’s been difficult for my entire team. We don’t have any answers to give folk. And we’re really, really conscious that we’re running out of time.”