BA canceled all flights from both airports Saturday, upending the travel plans of tens of thousands of people on a busy United Kingdom holiday weekend.
The airline also said that most long-haul flights due to land in London on Sunday were expected to arrive as normal.
The airline said a "high proportion" of short flights would operate Monday - but the Heathrow website showed a number of destinations affected by cancellations including Belfast, Dublin, Aberdeen, Manchester, Vienna and Copenhagen. At Heathrow, we operated virtually all our scheduled long-haul flights, though the knock-on effects of Saturday's disruption resulted in a reduced short-haul programme.
BA cancelled all Saturday afternoon and evening departures from Heathrow and Gatwick following what it called a "very severe disruption" worldwide to computer systems handling check-in, baggage sorting and reservations.
"We are extremely sorry for the inconvenience this is causing our customers during this busy holiday period", British Airways said in its statement.
Although the services resumed on 28 May but as many as 200 BA flights in and out of Hearthrow were cancelled on Sunday, according to a Guardian report.
Earlier on Monday, BA's chief executive said the outsourcing of jobs was not to blame for a "catastrophic" IT failure.
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British Airways says it is operating a full service out of Gatwick and a full long-haul service out of Heathrow today, following a worldwide computer system crash.
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 had been accusations of greed on the part of the airline from some, including the GMB trade union, who said that the problems which crippled the airline over the weekend could have been avoided if not for the outsourcing of many IT roles to India in 2016.
"We would never compromise the integrity and security of our IT systems", British Airways said.
The firm's CEO, Spaniard Alex Cruz, released a second video, after an initial one he released on Saturday, in which he apologized to customers for the "horrible experience" they have had to endure, going on to thank them for their "understanding and patience", Efe news reported on Sunday.
It advised passengers not to come to the two airports unless they have confirmed bookings.
The airline has suffered other IT glitches recently, leading to severe delays for passengers in July and September a year ago.