British Airways says it has fixed the IT glitch which caused more than 100 flights to be cancelled and more than 200 others to be delayed.
The airline said flights were returning to normal but warned that there may be "knock-on operational disruption".
At least 117 flights had been cancelled at Heathrow Airport, with 10 cancelled at Gatwick Airport.
The problem had caused BA to revert to using manual systems for check-in at airports, causing long queues.
It has apologised for the disruption and said customers on short-haul services from Heathrow, Gatwick and London City can rebook another day.
"We continue to advise customers to check ba.com for the latest flight information before coming to the airport, and to leave additional time," it said.
Major computer failure with British Airways at Gatwick this morning. This is the queue for the queue at check in. Currently stuck on ground cos pilot can't get data. Set off early if you have a flight pic.twitter.com/hL37TGRJ70— David Hewson (@david_hewson) August 7, 2019
The airline said the issue had not been a global problem, and involved two separate systems - one which deals with online check in, the other that deals with flight departures.
The issues were affecting flights across its network but not at every airport, it added.
Passengers have been invited to rebook flights on any other day up to next Tuesday.
Travellers have been expressing their frustration on social media.
Alex Brayson and his partner faced delays from Newcastle Airport en route to their wedding near Lake Bled in Slovenia on Saturday.
Despite boarding their first flight shortly before 06:00 BST, Mr Brayson said they missed their connecting flight to Venice, and their onward transfer was running three hours behind.
"We've got 25 of us flying to the wedding so we are growing anxious," he told BBC Newcastle.
"Me and my partner booked the trip so we've had to organise everything again."
Junior Great Britain wheelchair tennis player Abbie Breakwell, 16, is stuck in Heathrow with her mum after their flight and its replacement were both cancelled.
She is meant to be playing in a tournament in Belgium on Thursday - but has to register in person the night beforehand.
"They [BA] are trying to get us another flight but it's very hard at the moment.
"It's just a shame. I've been training for quite a while for these tournaments, but we can't go.
The teenager, from Nottingham, praised the "fantastic" airport staff for their efforts to get her to Belgium in time but added: "There's still hope, but it's slowly fading a bit."
Stuck at Heathrow airport due to @British_Airways technical issues, not sure I will make it to the Belguim tournament.Staff have been wonderful especially Dee Cassidy. #frustrated #lifeontour #wheelchairtennis @GranthamTennis @ITFWheelchair @PathtoSuccess1 @WChairTennisGB pic.twitter.com/w4o1oZhHdt— Abbie Breakwell (@AbbieBreakwell) August 7, 2019
Sam Angeli is stranded in Cyprus with his wife and two children after his flight home to London on Wednesday evening was cancelled.
He has been forced to spend more than £850 on two rooms at a hotel for the night.
Mr Angeli said his experience of BA had been an "absolute shambles".
"They've not emailed us, phoned us or messaged us about the delay," he said.
"We're just supposed to know what to do. I'm really angry with them."
Meanwhile, musicians heading to a four-day indie festival being staged on a boat are in a race against time to make it to Barcelona before it sets sail.
Belle and Sebastian singer Stuart Murdoch said he and 30 others - including Glaswegian outfit Camera Obscura - had planned to fly from Heathrow but were now trying to make a replacement flight at Gatwick.
@British_Airways 30 of us waiting to get onto a gate at Heathrow T 5. Heading to Barcelona, please hold the flight for us - BA0406 We are musicians flying to a festival, Belle and Sebastian, Teenage Fanclub, Camera Obscura. Please? Send a bus! 🚌— stuart murdoch (@nee_massey) August 7, 2019
At the scene
By Leigh Milner, BBC News, in Heathrow Terminal 5
There are lots of families travelling with children who are very annoyed, with many reporting that they can't get in touch with BA.
We've had people handing out bottles of water, anything that can help with the long wait. Passengers are being handed out leaflets about compensation.
They have been told to leave the airport and either go home or find a nearby hotel for which they will be reimbursed.
That's all to try and get those queues down.
This is not the first time BA has experienced IT problems, as it suffered major computer failures over the spring bank holiday weekend in May 2017.
On that occasion, the airline cancelled 726 flights and tens of thousands of passengers were left stranded.
It has been a problematic year for British Airways which is being threatened with strikes by pilots.
It is also set to be fined millions of pounds over a cyber attack on its security systems in which the personal data of up to 500,000 customers was stolen.
What rights do affected passengers have?
BA has a duty of care to passengers whose flights have been affected. Rights include:
- If the flight is cancelled, the airline must get them on an alternative flight to their destination. That might be a re-routed flight, perhaps including a stopover, or the next available direct flight. They can put customers on a flight with a different airline
- Some passengers will choose to take a refund instead, but the airline's duty of care ends at that moment. That means someone who takes the refund and books their own - more expensive - alternative flight is unlikely to have the difference reimbursed
- If a flight is still expected to depart, but late, then passengers will initially have to wait, but after a certain amount of time the airline should offer food, drink, and - if necessary - overnight accommodation. For delays of more than five hours, passengers get the same rights as they would if the flight was cancelled
In this case - when the disruption appears to be the fault of the airline - passengers may well be entitled to additional compensation under EU rules. This requires passengers to make a claim to the airline, and the level of payout depends on the type of flight and the length of delay.