Some quick reactions to what's rapidly turning into a kind of civil war, in no particular order.
1. Wagner has a difficult path ahead. Rostov is not that close to Moscow, and Russian forces have placed roadblocks along the main highway. So far, though, the regular Russian military hasn't put up a fight, and seems altogether unprepared.
2. Some level of apathy or support within the Russian military is key. Wagner doesn't have the logistics or ammunition to sustain a protracted campaign.
3. The Russian elite does not like Prigozhin. He's an outsider--a convict. Perhaps some of them have lost faith in Putin, but they have no reason to view Prigozhin as a suitable replacement. And I doubt he has the backing of the security services. I can't imagine any kind of Prigozhin regime would go very well or last very long.
4. But suppose Prigozhin's uprising is crushed--still the most likely outcome. It has left an indelible mark on Putin's government anyway. Throughout the war, Putin has retained his image as a stable protector. He has shielded the Russian public from Ukraine's battlefields: the Russian people can close their eyes and look away, while convicts and contract soldiers die.
No longer. The war has come to Russia: tanks are crawling across one of its largest cities. Putin can retain his strongman image by crushing Prigozhin with an iron fist--but he'll no longer be the "benevolent protector" who ended the chaos of the 90s. The chaos is back, and Putin can rule only by fear.
5. The present turmoil offers Ukraine a golden opportunity. If it lasts more than a few days, expect to see notable Ukrainian advances, perhaps in previously unlikely locations, like Bakhmut.
6. What happens to nuclear weapons? I have no idea. Frankly, I think Russian doctrine would allow or support a strike on Wagner. But I'm not sure anyone would carry out the order. I don't want Prigozhin in control of nuclear weapons, either, though. He's borderline crazy. From our standpoint, Putin is probably better than Prigozhin.
But something else odd: Prigozhin has spent his time attacking the military leadership, specifically Gerasimov and Shoigu. He rarely mentions Putin.