Airlines like BA have several significant problems when it comes to IT. First is that the underlying reservations systems were built when the only way to get the throughtput required was a mainframe. Even then, they used 7-bit ASCII which limited them to upper case and a small number of "special characters" (asterisks, dashes etc). They have managed to create XML based interfaces which is how web presentation occurs. What you don't see is quite how those XML interfaces work. You fill in a form and the result is an XML message. The recieving system(s) then have to complete several separate transactions to be able to send back the necessary response.
Reservations then require fares. Here the news is better in that the fare systems have largely migrated away from mainframes. However, distribution of fares between systems (so that airlines can sell interline tickets even within alliancs) remains batch based via people like SITA and ATPCO.
Ticketing has obviously evolved. In my lifetime we've gone from handwritten via printed to ATB to electronic but it's not all good news. Back in the days of handwritten you could string as many tickets together (ticekts had/have a maximum of 4 coupons/flight) to create a multi stop journey as you wanted. The trade press occasionally had pictures of travel agents with "daisy chains" of tickets strung across their office. These days the maximum number of coupons on an electronic ticket? 16 - Sixteen.
If we go back to the ATB we see another poroblem the airlines face - other airlines. As I said, tickets had/have 4 coupons per ticket number. But the ATB theoretically could handle 99 coupons and one major European airline built their system to take advantage of that. Every other airline had a problem. Why? Because the accounting systems assumed that no ticket number could be associated with more than 4 coupons.
Sticking with the "other airlines" theme they need to communicate with one another. Message formats need to be agreed across the industry (yes, you can have some bilateral agreements). One of my first ever IATA meetings was an annual affair. Because of some particular issues they decided to hold two meetings a year instead! Getting all the delegates together in one place at the same time isn't easy, yes the delegates probably travel on free staff tickets but IATA rules say the meetings have to rotate around the world so an airline that can easily send delegates to Geneva may have a problem getting to SIngapore simply because the planes are sull and staff travel is embargoed.
Departure Control Systems (DCS) (check in) are another area where issues arise. When electronic tickets appeared the DCS needed to be able to access the ticket database to check the passenger had paid and to mark the ticket coupon as used. But what happens if the flight then "goes tech" and the ticket needs to be reissued because the passenger has to get a connection to get to destination? You need to mark the coupon unused then go to reservations, make the new booking, then create a new ticket "in exchange" for the original, write a new ticket record, mark the original coupon used (again) and potentially another airline has to be able to access that ticket for the new connection.
Bottom line. There are multiple, interconnected systems, Changing one probably requires co-ordinated changes to at least one, and often several, other system.