Will the flight booking API tell me if the airline changes the flight time after I have issued the ticket?

Yes, most flight booking APIs will tell you, but it does not always reach you the moment the airline makes the change. When an airline moves a departure time, changes a flight number or drops a leg, that change goes on to the booking record held by the supplier. The API then exposes it either through a booking details call that shows the new timings against what you originally sold, or through a list of changed bookings that you are meant to read regularly.

What that means in practice is that your system has to go and ask. Very few suppliers push a message to you the second something moves, so the usual arrangement is a job that checks your live bookings on a schedule and flags anything that has shifted. Once it is flagged, your staff can decide whether the new timing still works for the customer, and your portal can email the customer about it automatically if you want it to.

Be honest with yourself about the gaps. Very small movements in the departure time are often not reported at all, a change made close to departure may reach your customer through the airline's own message before it reaches you, and some suppliers only report a change when you ask about that booking rather than in a list. A cancellation on the day of travel is still usually best sorted out by phone with the airline.

Ask whoever supplies your flight content which call returns schedule changes and how often they expect you to read it, then set your own checks to match that.