If the airline changes the departure time or cancels the flight, will the booking API let me know?

Yes, in most cases it will, but how quickly the news reaches you depends on the airline and the supplier you booked through. When an airline moves a departure time or cancels a service, that change is passed down to whoever holds the booking. A flight booking API can then show it against the booking on your side, so you can see the new time or the cancellation without having to check every file by hand.

What you cannot count on is instant notice. Some airlines send the change within minutes, others take a day or more, and a few of the low cost carriers prefer to write to the passenger directly rather than tell the agent who made the booking. Because of that gap, most agencies still take a quick look at their departures for the next few days rather than trusting the alert on its own.

What you are able to do about the change also varies. Sometimes you can accept the new time, ask for a different flight, or start a refund yourself through the same connection. Other times, particularly on complicated trips or with stricter airlines, the change has to be sorted out by the airline or the supplier, and all the connection can do is tell you that something has moved.

It is worth asking your supplier how these changes are sent to you, whether they arrive by email or appear on the booking itself, and which of them you are allowed to act on without calling anybody.