Let's assume, for a moment, that the thread of prophecy is not severed, and that we maintain some form of democratic government based on the US constitution. We keep having elections and whatnot.
I recognize that extremely bad stuff might happen that makes this not true, but if we have an overt authoritarian takeover, shooting in the streets, the French army liberates us or whatever, then talking about continuity of judicial appointments is a bit silly, isn't it? So, staying within bounds.