This isn't directed at anyone, but this is not about the American Civil War. I think I have established a good case about why this cannot be about America and would not involve them.
From what I can see, there are two clear blocks:
Prussian block:
Prussia (Franco-Prussian War)
Russia: Austrian involvement convinces them to seize the initative in the Balkans by weakening Austria- they smell weakness by Hungarian insurrection and the failure of the compromise of 1867 following Austrian involvement, because the Hungarians believe the war is a ploy to roll back the Hungarian compromise. Hoping to gain Galicia and a free hand in the Balkans.
Possible members:
Britain (I think this is unlikely, but some have argued otherwise. Prussian expansionism did not sit well with British public opinion. The Russo-Prussian alliance means that the English probably wouldn't join, as it would directly enable Russian expansion- after Crimea, it was paramount for the british to avoid this)
Italy (again, I think this is unlikely. They could use the war to seize irredentist claims against Austria and France. BUT they would probably sit neutral to try and extract concessions. Remember, they JUST unified Italy and were keen to garner international acceptance of their Rome annexation, by far the most important. They would not risk it all in a foreign war)
French Block
France (Franco-Prussian war)
Austria (In this timeline, let's say the Treaty of Prague was harsher and Austria was forced to give up Bohemia. Austrian opinion is outraged and they are looking to rectify their position and regain prominence within Germany and undo the humiliation)
Possible:
The Ottoman Empire- To stop an ascendant Russia in the Balkans
Britain- to safeguard the balance of power, both in Germany and in the Balkans (see several earlier posts)
Anyone vehemently disagree? Who would win?