No timeskipping yet, you nonce.  This is a delicate operation.  Before chapter 1 definitively goes anywhere, the characters I named are going with the watchers, and what happens after that will decide their fates.

Here's what happened: as Sho said, the destruction of the second seal caused a tremor and rifts.  Immediately some demons and angels arrived, who started battling, and over the course of hours as more arrived it escalated into a full war.  The humans started evacuation pretty quickly, and only after they were mostly out did the random skirmishes really coalesce into the full war.  During this time, you can have had your guys do whatever you want: leave, fight demons, try to take the abandoned Gilgamesh office down brick by brick, what have you.  I covered the time from the moment after the destruction of the seal through the hours.  

Everyone: aside from KYL, who's definitely with Souta, you can have had your characters do whatever you want during this time.  I outlined in my post a very brief overview of what the situation is looking like for the various factions.  The bottom line is: several hours after the destruction of the seal, every character that I named in the series is being approached by a different watcher.  I described the general procedure the encounter ought to take, but I'm leaving the specifics of the encounter, including the dialogue and the personality of your specific watcher, up to you.  In general, though the watchers are assholes.  I wanted to give you a little bit of freedom to have your character do something you like and is appropriate for them and then have you create the encounter with the watcher in a way you might enjoy as well.