Heathrow Airport has criticised Border Force after passengers faced "unacceptable queuing times".
Images on social media showed packed queues at the west London airport on Friday night, with travellers complaining of waits of several hours.
The Home Office, which has responsibility for the Border Force, has been approached for a comment.
Travellers at Heathrow used social media to complain about lengthy queues with inadequate ventilation, while some claimed people had fainted while they were waiting.
One Twitter user said: "There are people collapsing in Terminal 5 Arrivals due to overcrowding and no ventilation. Some Covid secure measures…"
Another said: "Horrendous kettling at Heathrow Terminal 5 arrivals tonight. No Border Force. No ventilation. No fire exits. No explanation. Held there for over an hour. People fainting. Panicked skeleton Heathrow ground team not knowing what to do."
On Saturday morning, Heathrow Airport tweeted that it was unable to say how long travellers would have to queue for immigration.
In a subsequent statement, the airport said it had "escalated" the issue with Border Force and expected it to provide a better service over the weekend.
"Border Force were aware of the extra demand from families and we are very disappointed that they did not provide sufficient resource.
"We have additional Heathrow colleagues to support in managing queues and to hand out passenger welfare including water, but we need every immigration desk to be staffed at peak times.
"We have escalated this with Border Force and expect them to provide a better service over the remainder of the weekend."
The UK's travelling restrictions were eased for many on 19 July when the government dropped the requirement for fully-vaccinated people and those under the age of 18 years and six months to quarantine after returning from amber list destinations.
Most countries - including the UK's most popular holiday destinations - are on the amber list.
Travellers coming from green list countries do not need to quarantine when they come back to the UK - whether or not they have had two doses of a coronavirus vaccine.
However, travellers from green and amber destinations must have one test before returning to Britain and another two days after they get back.
Only British nationals can enter the UK directly from a red list country, and they must pay to quarantine in government-approved hotels and be tested.
The travel industry has criticised the UK's changing travel rules, with airlines saying the system risked making international travel "more expensive, burdensome and uncertain compared to other countries".
Travel lists are currently updated every three weeks and the system is similar for England, Scotland, Wales and Northern Ireland.