Can we guess how long it will be until court procedings start or not? How will it work?
Paramount and CBS served the lawsuit on December 29. Axanar Productions and Peters have 21 days to file either an answer to the complaint or a motion to dismiss. I suspect they'll attempt the latter.
Within the next couple of weeks, we should see a motion from CBS and Paramount for a preliminary injunction to halt all production on the Axanar feature, as well as an order to pull distribution of Prelude and the four-minute Vulcan scene. Hard to say when the court will hear that but probably not before the end of this month at the earliest.