The answers to the OPs question are very varied depending on the specific individuals involved.
If you asked me what I think the most probable outcome is based on what I think is an "average" human template for all involved, I would say that the combined crew would face a choice between following the captain of the military ship or following the captain of the civilian ship. A certain number of the military crew would side with the civilian captain based on not having signed on to serve under military protocol for a 75 year stint. Then, if one side or another had a decisive and obvious advantage in numbers and resources, that would probably decide things peacefully - at least temporarily until and unless that situation ever shifted. The matter would be "tabled", with the commander of whichever side held the advantage running the show. But if they were approximately even - or if latter on, the situation did shift to bring things to a more even fight or even to the advantage of the opposed commander, then it would probably result in a fight. That might be decisive, or might lead to another temporary stalemate.
But, like I said: individuals. A well-known and respected captain like Picard might have the presence to unify the crew. Someone like Chakotay might have the wisdom to lead his crew to unify. A Captain Styles might piss off the civilians into trying to take power even if the odds were against them, or a captain like Esteban might not have the sense to relax the regs to allow the civilians a little breathing room, and force them to rebel - and his own crew, too, for that matter. Someone like Tom Riker (as the civilian captain) might have the knowledge and sneakiness to play nice and then steal the ship's resources necessary to rule the day... at least for a while. Maybe there's a specific popular lieutenant (on either side) who has been looking for a way to gain power and prestige, and now we have THREE factions (or more). Maybe the Starfleet crew has more members who sympathize with the Marquis cause - maybe less, maybe none, maybe some of them HATE the Marquis for some reason. Maybe the Marquis crew don't like their captain for whatever reason and were only following his orders because of the Marquis command structure that is now 75 light years away. And on and on.