The airline said it "may take some time" to reunite travellers with their bags.
"Our IT systems are now back up and running and we will be operating a full flight schedule at Heathrow and Gatwick," the British Airways (BA) said.
Embattled Chief Executive of BA Alex Cruz has blamed a power surge for the disruption, which affected 75,000 people.
He had ruled out resigning over the crippling flight disruption and maintained that the computer glitch had nothing to do with cutting costs or outsourcing IT services to India.
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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The union spokesperson said it could have been avoided had "hundreds of dedicated and loyal" not been replaced by cheaper Indian staff in 2016.
Cruz also said that no BA passengers' data had been compromised in the IT meltdown and said there was no evidence it was the result of a cyber attack, promising not to allow such an outage to happen again.
Cruz said: "We have no evidence whatsoever that there was any cyber attack of any sort".
Cruz said he was "profusely sorry" to the thousands of passengers still stranded at airports worldwide.
Heathrow, Europe's busiest airport, advised affected BA passengers not to travel to the airport unless their flights had been rebooked, or were scheduled to take off yesterday.
BA is liable to reimburse thousands of passengers for refreshments and hotel expenses, and travel industry experts have suggested the cost to the company - part of Europe's largest airline group IAG - could run into tens of millions of pounds.
In a statement yesterday, the airline apologised again for the "frustration and inconvenience" to passengers and said it was continuing to get people reunited with their luggage "as quickly as possible".
The airline also admitted that it was aware some passengers were unable to file a delayed bag report on its website.
Thousands of BA passengers worldwide are still without their bags and have been bombarding the airline's lost luggage phone lines, website and social media feeds.
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