The airline was hit by a worldwide computer system power failure on Saturday, causing cancellations and delays for thousands of passengers.
All long-haul services will fly from Heathrow, but with delays, BA said.
The airline urged people to check the status of flights before before travelling to the airport.
Heathrow Airport is providing an updated schedule on its website, as is Gatwick, where Sunday departures are delayed but not cancelled.
The BA hotline - which passengers were advised to call - was reportedly down on Sunday afternoon.
BA told people not to arrive at the airport too early, as people are not being admitted into Terminal 5 until 90 minutes before departure.
The airline apologised to customers for the issue, which is thought to have been caused by a problem with the IT system's power supply.
In a statement released on Sunday, chief executive Alex Cruz said: "I know this has been a horrible time for customers. We're not there yet, but we are doing our very best to sort things out for you."
The airline is liable to reimburse thousands of passengers for refreshments and hotel costs.
Customers displaced by flight cancellations can claim up to £200 a day for a room (based on two people sharing), £50 for transport between the hotel and airport, and £25 a day per adult for meals and refreshments.
One traveller from Seattle said she had spent the evening in a sleeping on the floor of a hotel conference room.
Ashley Tracey, who is trying to get to Mumbai for her friend's wedding, said she had been queuing to rebook her flight for six hours.
She said: "There's no information I can't seem to get through online, I don't live here so I don't have a phone that works here."
There have been reports that some passengers who departed from Heathrow on Saturday found their luggage was not at their destination when they landed.
Terry Page, 28, arrived in Fort Worth, Texas after delays and said "about 50" passengers did not have their check-in luggage.
Passengers slept on yoga mats, handed out by the airline
Thousands of bags remain at Heathrow Airport, but BA has advised passengers not to return to collect them, saying they will be couriered to customers.
The airport also said passengers whose flights were cancelled should not travel to Heathrow unless they have already rebooked their flight.
A spokesman said staff were "continuing to give passengers free water and snacks".
Some passengers at Heathrow Terminal 5 have been waiting for six hours, since returning to the terminal to rebook flights
Some flights at Gatwick Airport are delayed, but no flights were cancelled on Sunday morning
On Saturday passengers described "chaotic" scenes at the airports, with some criticising BA for a lack of information.
The airline apologised and said it was refunding and rebooking customers.
The company's 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.
Queues for Heathrow's Terminal 5 stretched out of the building on Sunday
The IT failure affected check-in and operational systems, including customer service phone lines.
BA said although some of its IT systems had returned, "there will be some knock-on disruption to our schedules as aircraft and crews are out of position around the world.
"We are repositioning some aircraft during the night to enable us to operate as much of our schedule as possible throughout Sunday."
A spokesman added: "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."
The GMB union had suggested the failure could have been avoided, had the airline not outsourced its IT work.
BA denied the claim, saying: "We would never compromise the integrity and security of our IT systems".
Passengers were left stranded outside Heathrow Terminal 5 on Saturday as a result of the IT failure
EU flight delay rights
- If your flight departed the European Union or was with a European airline, you might have rights under EU law to claim if the delay or cancellation was within the airline's control
- Short-haul flights: 250 euros for delays of more than three hours
- Medium-haul flights: 400 euros for delays of more than three hours
- Long-haul flights: 300 euros for delays of between three and four hours; and 600 euros for delays of more than four hours
- If your flight's delayed for two or more hours the airline must offer food and drink, access to phone calls and emails, and accommodation if you're delayed overnight - including transfers between the airport and the hotel