Heathrow Airport is experiencing waits of three and sometimes six hours at border control, Emma Gilthorpe, its chief operating officer, told a home affairs select committee on Tuesday.
Passengers this month complained of having to wait up to seven hours, with one traveller calling it "inhumane".
Entry is controlled by the Border Force, managed by the Home Office.
Meanwhile, hundreds of staff who work at Heathrow passport control are set to take industrial action in a dispute over duty roster changes.
Unions representing Border Force officials said the delays were partly caused by Covid-19 restrictions, requiring immigration officials to work in a bubble of 10.
They said this prevented more staff from being stationed if the border was particularly busy.
Ms Gilthorpe said that before the pandemic, the queues for EU arrivals were supposed to be 25 minutes, and 45 minutes for non-EU arrivals.
She said she expected waiting times to increase during the pandemic but the current levels were "unacceptable".
"We are seeing significant pressure on the border and we are seeing very long queues, and that is a worry," Ms Gilthorpe said.
She said more resources had been introduced but the problems were caused by "the level of complexity of the processes" and the way those resources were being used.
International arrivals must now provide proof of a negative Covid-19 test result from 72 hours before departure to England.
The must fill out a passenger locator form that contains booking details for a hotel quarantine stay, or two test kits for at home quarantine.
"It is deeply frustrating as the operator of the airport when you have a queue full of people and you only have two desks open," Ms Gilthorpe said.
"It is rare to see all the desks manned and we have to find our way to how we make that happen so we can get that flow."
Heathrow's self-service e-gates at passport control are closed.
Ms Gilthorpe hoped that by the summer, the gates would also be able to scan locator forms and pre-departure testing results, but that had not yet been integrated into the system.
She said that if the long queues continued after international travel is allowed, it could endanger the country's economic resurgence.
"Heathrow is Britain's hub airport, it is the front door," Ms Gilthorpe said.
"We have to get our economy moving and we have to make sure we are capable of receiving people.
"If you have a poor experience at the border, there's a risk you won't come back again.
"That traffic will go to Charles de Gaulle, it will go to Frankfurt and we will miss out on that economic resurgence."
Staff in the PCS union served notice of industrial action over rosters they describe as "unworkable".
This will involve a month of action short of a strike by hundreds of staff, including working to rule and a ban on overtime.
Under the roster changes, staff will no longer be able to swap or request certain shifts, the union said.
In February, staff voted for strike action over the changes, the union said.