Passengers also face the threat of future disruption after BA pilots voted to take industrial action over a pay dispute on Wednesday.
At least 117 flights were cancelled across Heathrow, Gatwick and London City airports on Wednesday, affecting tens of thousands of passengers.
BA could be forced to foot a compensation bill worth more than £8million if all those affected claim under European Union rules.
The airline said in a statement: “We have resolved the temporary systems issue from earlier today and apologised to customers who were affected.
“Any customers whose flights have been cancelled have either been rebooked or offered a refund.
“We plan to operate our normal schedule tomorrow, however there may be some knock-on disruption.
Video report by ITV News Correspondent Lucy Watson
“We continue to ask customers to check ba.com before heading to the airport to get the latest status of their flight.”
The airline said it appreciated "how frustrating their (customers') experience had been" and that BA teams worked "tirelessly" to help travellers.
At Heathrow, there were long queues of passengers and an error message on the BA app as services lagged more than five hours behind schedule.
The airline was forced to use a back-up manual system in a bid to cope with the problem.
Darren Rowe, from the Cotswolds, said his 10.20am flight to Hamburg from Heathrow for business meetings was cancelled before “all chaos let loose”.
He said: “There were massive queues, it was queue here, queue there, nobody was saying anything. The lack of information was just pathetic.
“You’ve got young families in that queue, people going to weddings, birthdays, on business. They could have had somebody come around with water updating people about what was going on.”
BA’s IT problems come after it suffered a major computer failure over the spring bank holiday weekend in May 2017, stranding tens of thousands of passengers and costing owner AIG around £80 million.