Border Force staff are struggling to cope with queues to check passports at Heathrow due to stress, a whistleblower claims.
Target times for some passengers to clear immigration were missed in the last two months, new figures show.
Extra staff have been redeployed from Channel ports to help airport colleagues deal with delays.
The Home Office said it had an "agile, flexible and intelligence-led workforce" to respond to needs. A spokesman said Border Force made "every effort" to minimise delays.
But he added: "When very large amounts of passengers arrive in a short space of time, it can mean a longer wait while essential border security checks are conducted."
A long-serving member of south-east Border Force staff, who spoke to the BBC on condition of anonymity, said: "We are being crucified. There is a high percentage of long term sick due to stress. We are being pushed completely and we cannot cope."
He claimed some Border Force staff had to start their shift at Dover, drive to Heathrow to do three hours' airport work, then drive back.
Kevin Mills, of the Public and Commercial Services Union, said: "Redeploying staff from anywhere, particularly the south east when we've seen unprecedented levels of clandestine activity trying to cross the Channel, really isn't a good move."
A Home Office spokesman said: "Security is not compromised when staff are deployed from other ports or airports, it is maintained.
"Border Force has been set up to be agile, flexible and intelligence-led, deploying quickly according to need and responding to a variety of demands."
Heathrow, which monitors immigration queues, said Border Force had a target of processing 95% of non-European Economic Area passengers within 45 minutes.
The number fell in June to 89.4% in Terminal 2, 91.3% in Terminal 3, 93.3% in Terminal 4 and 94.1% in Terminal 5. Three of the four terminals also missed the 95% target in May.
An airport spokesman said: "Heathrow has been talking to Border Force about how they can improve immigration waiting times during peak periods.
"There is not a trade-off between strong border security and a good passenger experience - at the UK's global gateway we want them to do their part to deliver both."
Tony Smith, a former head of the Border Force, said airlines needed to help.
"What we don't control in government is the aircraft coming in, what times they arrive, how many people are on them. So it does require some input from the airports and the airlines about how they can help."
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