The ICO plans to impose a penalty of 1.5% of the airline’s annual turnover.
The flight was diverted after it failed to land into Gibraltar.
Strong winds caused by Storm Erik are believed to be the cause of the flight reattemping landing.
The BOAC livery will remain on the Boeing 747 until the aircraft is taken out of service in 2023.
Britain's largest airport said early on Wednesday morning that a lighting issue had affected both runways.
BA is contacting two groups of customers not previously notified.
The Information Commissioner’s Office has started making inquiries after customer data was hacked.
The airline said it was investigating the breach “as a matter of urgency”.
British Airways is contacting customers affected by a data breach that saw around 380,000 payments compromised.
The stolen data did not include travel or passport details.