I'm curious how this mechanic works since I've never actively tried to use it. Does it Allow any battle on a system to be replayed, or one that happened that turn? If boarding pods are in play, do you duplicate the ships taken? Can anyone replay the battle, or just the riftborn? Can it only be replayed once? And does it take into account Juggernauts that are in 'cooldown' mode? 

Anyone got answers? 

There is additional button in post-battle screen that allows you to redo it and change last-minute preparations, like lane composition and tactics, or retreat instead of engaging. It is quite useful sometimes.

So yeah, it can only affect battles from one turn, as notifications are cleared on turn change.

First battle resolution is undone as it never happened, so captured ships are not duplicated and instead returned to their initial owner.

I'm not sure if anyone besides owner of the singularity is allowed to replay the battle. But it can only be replayed once.

Jaggernaut should be reset to the state it was pre-battle, just like any other ship involved.