Computer problems grounded scores of planes over the weekend, leaving thousands stranded at airports.
BA said it intended to run a full long-haul schedule at Heathrow and a “high proportion” of its short-haul programme, and a full schedule at Gatwick.
It claimed it was making “good progress” in recovering from the worldwide IT glitch.
But passengers are still being warned to check before travelling to airports and a Heathrow spokeswoman said there was still “some disruption”.
Customers have been left queuing for hours in packed terminals over the last few days and some had to bed down on terminal floors on Saturday.
Many complained of scant information from staff.
Experts predict BA is facing a huge compensation bill, estimated at more than £100m.
The GMB union has claimed the problems were down to BA cutting “hundreds of dedicated and loyal” IT staff and contracting the work out to India to save money.
Mick Rix, national officer for aviation at the union, said: “This could have all been avoided.
“In 2016 BA made hundreds of dedicated and loyal IT staff redundant and outsourced the work to India.
“BA have made substantial profits for a number of years, and many viewed the company’s actions as just plain greedy.”
The redundancies caused protests and union outrage, with BA saying at the time that it had signed a contract with India’s Tata Computer Services “to be the supplier of some IT activities”.
Around 200 staff were affected.
The airline has so far given little detail on the cause of the problems, only saying a “power supply issue” was to blame.
A BA spokeswoman said operations had started to return to normal by Sunday.
She said: “We operated a full schedule at Gatwick on Sunday. 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.”
Alex Cruz, the airline’s chairman and chief executive, issued a video message on Twitter to apologise and reassure passengers about their lost luggage.
He acknowledged that it had been “a horrible time for customers” and apologised for “these very trying experiences”.
The IT outage also had a knock-on effect on BA services around the world, while passengers who did get onto flights from the UK reported arriving without luggage.
Video has emerged of a BA employee at Venice Airport threatening to call the police on a woman who asked about the policy on customers who do not have money to pay for hotel rooms.
Stacy Irish, who posted the footage on Twitter on 28 May, said: “I was told it has nothing to with BA if customers can’t afford it. She then said she would call the police.”
Some experts expect the disruption to linger for several days, as planes and aircrew are returned to their positions and the backlog of passengers is cleared.
Air industry consultant John Strickland said: “There’s a massive knock-on effect.
“Customers and – from the airline’s point of view – manpower, dealing with the backlog of aircraft out of position, parking spaces for the aircraft… It’s a challenge and a choreographic nightmare.”
In 2016, an electrical fault at the headquarters of American airline Delta grounded more than 1,500 flights.
Delta was criticised for its “cobbled” IT infrastructure, which was described as a problem for many airlines.