The first flight carrying British nationals from Afghanistan arrived at RAF Brize Norton in Oxfordshire on Sunday night.
Embassy staff were among those on board the RAF Voyager.
The rescue efforts are part of Operation Pitting, the codename for the military support to the evacuation of British nationals and former British staff eligible for relocation under the Afghan Relocation and Assistance Policy (ARAP).
About 600 troops from 16 Air Assault Brigade have been deployed to Kabul to work with US forces to secure the airport, so flights can continue as Afghans and foreigners scramble to leave.
Defence Secretary Ben Wallace told BBC Breakfast: "We should have capacity for over 1,000 people a day to exit to the UK.
"Currently this is not about capacity on planes, it's about processing speed, so that's why we're trying to fix that."
Mr Wallace added: "Our timescale which we'd originally planned, which was August 31, will be out.
"We will try our very best to get everyone out, it's one of the biggest regrets of the speed of the collapse of the Afghan government is that those timetables will, no doubt, have to be shortened and that's what we're moving extra assets to do…"
He added: "We put in over 600 forces yesterday, today and over the weekend, to make sure that we can keep a secure part of the airport functioning and, at the same time, to effectively process, manage and escort people on to our flights to get them out of Afghanistan."
Mr Wallace said the Government was aiming to fly out a further 1,500 people over the next 24 to 36 hours or slightly longer.
Kabul airport has so far not come under attack but there are fears that could change quickly with Taliban insurgents now effectively in control of the capital.
Footage of people climbing onto planes at the airport to leave the country has gone viral on social media.