Sunday departures from Gatwick, London's other major airport, are delayed, but none have been cancelled. The airline urged people to check the status of flights before travelling to the airport.
British Airways had earlier said it plans to resume most of its flights out of Heathrow and Gatwick airports today following major IT system disruption that had grounded planes yesterday.
A BA spokesperson added: "We are continuing to work hard to restore all of our IT systems. We are extremely sorry for the huge disruption caused to customers throughout Saturday and understand how frustrating their experiences will have been.
"We are refunding or rebooking customers who suffered cancellations on to new services as quickly as possible and have also introduced more flexible rebooking policies for anyone due to travel on Sunday and Monday who no longer wishes to fly to/from Heathrow or Gatwick."
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"We would never compromise the integrity and security of our IT systems," BA said in reference to GMB union's claim.
BA chief executive Alex Cruz had said it was believed "the root cause [of the computer problems] was a power supply issue".
The company said there was no evidence the failure was the result of a cyber attack.
The airline is expected to have to pay out huge sums in compensation, including the cost of hotels, transport and meal expenses for stranded passengers.
"This could have all been avoided. In 2016 BA made hundreds of dedicated and loyal IT staff redundant and outsourced the work to India," said Mick Rix, national officer for aviation at GMB.
"BA have made substantial profits for a number of years, and many viewed the company's actions as just plain greedy," he said.
The move in 2016 had sparked protests and outrage from members of the union.
Thousands of bags remain at Heathrow Airport, but BA has advised passengers not to return to collect them as they will instead be couriered to customers.
BA said although some of its IT systems have returned, "there will be some knock-on disruption to our schedules as aircraft and crews are out of position around the world".
Delays were also reported in Rome, Prague, Milan, Stockholm and Malaga due to the system failure, which coincided with a bank holiday weekend and the start of the half-term school holidays for many people in the UK.