Kirk would have been revealed to have slept with a prophet during one of his trips to the past and thus conceived his own ancestor. Not to mention Kira's totally tapped.
The trial would be held in a 21st century nudist court which would inevitably end with Q trying to seduce Janeway as a "test"
Broken Bow would be a one parter with Picard since you could cut out all the Vulcan racism.
And since you forgot him... Archer would be lost in the Delta Quadrant never to be heard from again.