Taking Rostov-On-Don also takes the entire Russian military effort in Ukraine hostage, that now is the major supply base for almost all Russian force in Ukraine, if Prigozhin took the Southern Command (no way to verify if he had done it) which mean nothing is going out to Russian troop in Ukraine unless Prigozhin said so.
Depends on what Prigozhin want to do and whether or not Putin will relent. Wagner may hold out in ROD for a long period of time, and either Putin would have to charm Prigozhin or have to retake ROD by force, which with majority of Russian combat power already in Ukraine, I don't know how it would be possible, you can't send the National Guard and FSB unit to fight an offensive on a battle hardened Wagner. So, I would bet this is going to be peaceful way out, but how long does it take and how much damage to Russian force in Ukraine is going to be unknown, I suppose Prigozhin won't release any supply until his demand is met. I mean, he ain't stupid, he wouldn't take ROD and let the supply free float into Russian military hand.