The airline was forced to revert to manual check-in for some flights, BBC reported.
In a statement the airline said: "We are experiencing some systems problems this morning which are affecting check-in and flight departures".
The cancellations come as BA passengers posted images on social media that showed long queues of holidaymakers as well as error messages with the BA app.
"We are very sorry to our customers for the disruption to their travel plans", the airline said.
BA has cancelled more than 80 short-haul flights from Heathrow and a further 10 from Gatwick so far today, so it might not be an exaggeration to say that hundreds of flights may end up in the bin today.
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Anyone who may be affected by the issues with British Airways can check their flight status here.
Southern said the fires had been put out and lines reopened but residual delays are expected to last until about 14:30 BST.
He told PA: "Most flights are now either fully booked or have shot up in price or involve stopovers. Little stressed at the moment, especially as I am flying over to get married".
"We strive to give passengers excellent service and quality information. We apologise for any inconvenience".
"It was only after Network Rail investigated and then closed all four tracks that we could replan our services and update passengers with accurate information".
The airline is also set to be fined 183 million pounds ($222 million) over a cyberattack on its security systems a year ago in which the personal data of up to 500,000 customers was stolen.