In DLM canon, when they have a whole lot deaths happening at once, the reapers are allowed an 8-hour window to get to the dead and send their souls on to a central processing location where the supervising reaper sends them on to their lights.

I kind of tweaked that a little bit in this story, but it's one of those weird things where I didn't have to make up something out of whole cloth for how the reapers would operate during an apocalypse.