The airline is "closer to full operational capacity" after an IT power cut resulted in mass flight cancellations at Heathrow and Gatwick.
Thousands of passengers remain displaced, with large numbers sleeping overnight in terminals.
The British Airways has not explained the cause of the power problem. So far today, 13 short-haul flights at Heathrow, Europe's busiest, have been cancelled.
The BA is liable to reimburse thousands of passengers for refreshments and hotel expenses, and travel industry commentators have suggested the cost to the company - part of Europe's largest airline group IAG - could run into tens of millions of pounds.
Customers displaced by flight cancellations can claim up to 200 pounds a day for a room, 50 pounds for transport between the hotel and airport, and 25 pounds a day per adult for meals and refreshments, the BBC reported.
Passengers on cancelled flights have been told to use the BA website to rebook.
Chief executive Alex Cruz has posted videos on Twitter apologising for what he called a "horrible time for passengers".
But no-one from the airline has been made available to answer questions about the system crash, and it has not explained why there was no back-up system in place, the BBC reported.
Heathrow advised affected BA passengers not to travel to the airport unless their flights had been rebooked, or were scheduled to take off today.
Thousands of passengers had their travel plans disrupted as a result of the problem, with one workers' union blaming the outsourcing of IT jobs to India for the chaos, a claim denied by the airline.
BA's GMB union has said outsourcing IT jobs to India could have made the problems worse.
The union spokesperson said it could have been avoided had "hundreds of dedicated and loyal" not been replaced by cheaper Indian staff in 2016.
The Sun newspaper, quoting one source close to the airline said the problems could have been limited had IT staff outsourced to India known how to get its back-up system online quickly.
They added: "The third-party support providers are generally quite inexperienced meaning it makes a situation like this difficult to recover."
Meanwhile, the BA said there was no evidence the failure was the result of a cyber attack.
Experts predict the knock-on effect on the BA could continue for several days.
Passengers faced hours-long lines to check in, reclaim lost luggage or rebook flights at Terminal 5, BA's hub at Heathrow.
BA operates hundreds of flights from Heathrow and Gatwick on a typical day - and both are major hubs for worldwide travel.
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