BA grounded all planes from Gatwick and Heathrow on Saturday, causing chaos for thousands at the start of the school half-term and bank holiday weekend.
The disruption has continued into Sunday, with dozens of Heathrow services cancelled and passengers warned not to go to the airport without rebooking or checking their flight status.
Some shops at the airport had run out of food and many people slept on the floor on mats and blankets, the Press Association has been told.
Welsh international table tennis player Chloe Thomas arrived on Sunday around four hours before her 7.30am flight to Germany for the World Table Tennis Championship in Dusseldorf.
She said there were "just queues everywhere" and her plane was cancelled at the last minute.
"We stood in the check-in queue, not moving, for about an hour then it came up on the screen that the flight was cancelled," she said.
"To be honest I wasn't surprised. We didn't think we would make the flight because we were in the queue for such a long time."
After finding out they would not be departing for Dusseldorf as planned, the group joined "another queue the length of the airport" to get a number to rebook.
"It's chaos, people are running about all over the place trying to rebook," Thomas said.
"There's no-one to help, no leadership, it's just mental. There are lots of people everywhere.There's nowhere to sit, so people are just lying on the floor, sleeping on yoga mats."
Airport staff had handed out the yoga mats, as well as thin blankets, for people who were stuck there overnight, she added.
The problem has been caused by a worldwide systems failure, which BA said is believed to have been caused by a power supply issue.
Thomas also said one of the shops has already sold out of food.
She and her teammates, Callum Evans and Daniel O'Connell, have been booked onto another flight, but now need to get to Birmingham.
London's Kings Cross railway station was packed with people trying to make their way north, with travellers queuing out of the doors and onto the platform for the 9am train to Edinburgh.
Those that could not get on ran to the doors of the next service at 9.30am when the platform was announced, but were met on boarding with a warning from the train guard that seats could not be guaranteed for all passengers.
Passengers lucky enough to be aboard one of the few flights taking off on Saturday later found their hold luggage had not made it onto the plane with them.
Terry Page, 28, from London, flew from Heathrow to Fort Worth, Texas, where he and "about 50" others were told they would have to wait until Monday before being reunited with their bags, he told the Press Association.
As IT teams tried to fix the system:
:: Travellers have been told to check the airline's website and Twitter account for updates before setting off for Gatwick or Heathrow.
:: All of BA's check-in and operational systems were affected by the issue, including the airline's customer service phone lines and rebooking function.
:: The incident had a knock-on effect on BA's operations around the world.
:: Customers whose flights were cancelled are being refunded or rebooked on to new services and other options are available for those who no longer want to fly.
:: BA chief executive Alex Cruz said the airline was "extremely sorry" for the "huge inconvenience" suffered by customers, especially families heading on half term holidays.
BA said it is aiming to operate a "near normal schedule" at Gatwick and the "majority of services" from Heathrow.
There were dozens of cancellations at Heathrow on Sunday morning and delays of around 30 minutes to their flights from Gatwick.
The airline was unable to say how many flights would be cancelled or how long the disruption is likely to continue for.
A spokesman said: "Our focus is on updating customers and doing what we can to get them to their destinations as quickly as possible."
Experts predict the knock-on effect could continue for several days and BA is facing huge compensation costs.
The glitch is believed to have been caused by a "power supply issue" and there is no evidence of a cyber attack, the airline said.
There were issues with its online check-in systems in September and July last year, causing severe delays for passengers.
Mr Cruz said in a statement on Sunday afternoon that many of the systems are back up but the knock-on effect is continuing.
He warned passengers that the BA Heathrow terminals were "very congested" and that people would not be let into Terminal 5 until 90 minutes before the scheduled departure time of their flights.
Anyone who had their plane cancelled can rebook until the end of November or have a full refund, he added.
He apologised again for the "horrible time" and "very trying experiences" customers have been through.