The bigger point to be made here, I think, is this: Any Fox affiliate whose news is produced by another station with an 11 PM newscast (or vice versa) already has a contingency plan for what happens when Fox programming runs past 10 - and they've had to use it plenty of times, especially during MLB postseason.
In many cases, the plan involves starting to tape a newscast once the delay runs past a certain time, so if the network is still going at 10:20, for instance, they'll start taping a :30 local show. If Fox signs off at any point between then and 11:35, that :30 taped show will start playing. If Fox goes past 11:35, some stations will start a live local newscast after that; others pack it in for the night and leave the tape to run.
The only difference between that and this is that for this January late night, the stations already know exactly how late the network will be and can adjust accordingly. Depending on the station, that might mean taping a Fox newscast to run at 11, or simulcasting on both stations at 11, or even taping a show for the ABC/NBC/CBS station and running live at 11 on Fox. It's just another one-time delay.
