Hmm, things could go in many ways, but I think that Nice will only get support if they show that they're more than just a splinter faction. At the moment Paris still controls the most important regions of the country, but right now a lot depends on the loyalty of the army, and specifically the body of the soldiers. The officers might lean to the right, but large left wing sympathies amongst the common soldiery could still play a major part. The industry will support Paris, they are the core of the left.
Great Britain could lean either way here, especially because Nice started the civil war, it would be hard to sell the British public on going to war, or supporting forces against the legitimate government of France. If they would help Nice, Labour would be in an uproar, and there could be crippling strikes and in extreme circumstances even revolutionary sentiments spreading throughout the country.
The Soviet Union would certainly supply Paris. Stalin want the West destabilized, and a civil war in the heart of Western Europe certainly achieves that goal. If Germany gets properly involved, he might even start to move westwards, taking Poland and the Baltic states.
Germany could lean either way I think. Or true neutrality of course. The Hitler regime could be associated with the Spanish civil war, and that might influence the decission to not intervene. Moving in to 'protect' certain regions of course would be no more than expected. And that could lead to deeper and deeper involvement, which could actually draw in the British as well, even on different sides.
Franco's Spain could give some support to Nice, but knowing him it's also possible that he keeps his hands off the whole thing. But I think that some support is likely, even if it doesn't mean all that much considering his resouces.
The Italians have been suppressed, but if they think that Germany and Britain are sufficiently distracted, they could make a move on Corsica for instance. But the biggest lure would be the colonies, especially when the civil war escalates further you could see them fighting amongst each other, and countries being given some of the less important of them in trade for support.
I presume that the Americans wouldn't really get invovled at all, at least not in an early stage. Isolationism still is a big thing after all.