Other flights were delayed as passengers reported long queues at the London airport’s Terminal 5 on Wednesday evening.
One traveller said: “They just said to me ‘stay in the queue’. It’s not terribly helpful… It’s complete chaos.”
Another said: “We’re really not getting any information at the moment. Nobody’s being sorted out… They can’t re-book the flights because they’ve got no computer system. Everyone’s just sitting here.”
Hundreds of passengers were told to return on Thursday.
The airline said it had experienced a “systems issue”, leading to a backlog of flights and some cancellations.
On Thursday morning, a BA spokesperson said: “Our flights are operating today, however we expect some knock on disruption from yesterday’s combination of a temporary closure of Heathrow Airport’s air traffic control tower and a supplier system issue which affected a number of airlines.
“We are doing all we can to keep disruption to our services to a minimum, and are sorry for the disruption to our customers’ travel plans.”
On Wednesday, a number of airlines suffered a separate set of delays caused by a fire alarm at Heathrow’s air traffic control tower, forcing some flights to be diverted to other airports.
The alert affected BA services at Terminal 5.
A Heathrow Airport spokesman said the alarm “impacted operation of the airfield for a short while”.
The disruption comes as airports across the country gear up for the busiest part of the year as the summer holiday season begins in earnest.