Manually. The TPF reservation systems have a concept of "Queues". They would place travel records that needed to be reaccommodated onto a queue. Then, reservations agents would "work the queues" from their green screens, make phone calls, etc.
>how is it handled today given all the intertwined applications
Depends on the airline, and the system, but the general answer is "partially automated". The processes for less widespread issues is more automated. Think like a major storm in the northeast. Global outages are less automated because you're dealing with multiple issues, not just passengers.
IBM offered ACP (Airline Control Program) with their new mainframes running OS/360 in 1965. Later they changed its name to ALCS and TPF, still used today by reservation systems.