The airline announced that it had cancelled all flights for the rest of Saturday after passengers spent hours waiting to see if their flight would take off.
One passenger, Claire Dancer, from Swindon, who was waiting for a flight from Heathrow to Corfu, said she was infuriated by a lack of information.
She told Sky News: “We arrived at the airport at 9am. We queued for three hours just to drop off our luggage. No information was given.
“We were then pulled from the queue and told to run to the gate for our flight otherwise we would miss our flight.
“We arrived at the gate having had no time to visit any utilities only to be told there was a delay. But there was no information. People at the gate were really cagey.
“My daughter found out on BA Twitter that the flights were cancelled before six. We realised nothing was going to happen.
“The pandemonium increased as the airport merged into a massive queue. Nobody knew where to go, what to do and were just told we would have to leave the airport but were unable to reclaim our luggage.
“We have been given no information from anybody. We can’t get through to BA, everything is just disconnected.”
Another passenger tweeted: “Still on the tarmac at Leeds. #britishairways reckon Heathrow is so backed up so we can’t set off. No way we’ll make our Vegas flight.”
The chaos was not confined to the UK, another passenger described the situation in Berlin.
Simon Bucks told Sky News: “My wife and I are stranded in Berlin because of the BA computer crash. They are not booking anyone into hotels – we have had to find one ourselves.
“They have no idea when we can get a flight out because the system is still down. It is chaos. The BA office in Berlin is only open Monday to Friday.”
Linda Murakami, 55, had flown in from Pennsylvania, US, for a connecting flight to Naples, Italy, only to find it had been cancelled.
She said she would be cancelling a “dream holiday” to the Amalfi coast with her family.
Ms Murakami, who retired early from a pharmaceutical company, said: “This trip was my retirement present.
“There are a lot of other things that can cause disappointment and could put everyone in a worse situation.
“We just have to be flexible. We have travel insurance so feel that because we would miss part of the vacation, we might as cancel it and just go again at a different time. For me, it is an inconvenience.”
Air travel experts said BA could face a massive cost in lost revenue and compensation for affected passengers.
Malcolm Ginsberg, editor in chief at Business Travel News, said: “There is no question – the EU denied-boarding regulations will have to apply.
“They have broken all the rules and they will have to deal with it – it’s going to be a very expensive situation for BA.”
Airlines must also provide food and drink if their passengers are delayed by more than two hours.
Civil Aviation Authority guidance states that anyone who is more than three hours delayed arriving at their destination could be entitled to compensation.