Ward Moore's Bring the Jubilee had the victorious Confederacy support an alt-Germany in an alt-Great War before siding against them in an alt-Cold War. The U.S. was a semi-independent, disarmed "protectorate" of the South that at least the main character suspected would be the battlefield if Germany and the South went to war.
While it may not start in 1914 and may not play out with exactly the same sides or in any way resembling the unending clusterhump that we got in OTL, I think some form of Great European War is likely to happen between the unification of Germany and the early 20th Century. I would say that though not yet a done deal, Unification was on its way towards happening and would be unlikely to be butterflied away. Once there is a German Empire straddling Middle Europe, the balance of power that kept the peace after Napoleon is gone. Without the balance of power, the great European sport of war is likely to follow.
I can see the U.S. and the C.S. reacting in one of several ways. I would say that my guess would be neutrality for both sides. Unless there was an active desire for revenge on one side, I would imagine that both countries would not want to get into war. Now if either the C.S. felt it owed a debt the Britain or France or the U.S. owed those countries payback for their intervention, I could see just that one country entering. Obviously if they both felt that they "owed" the Entente, then an American Front is possible--though almost certainly without trench warfare outside of sieges of major cities.
I can't say how likely it is, but if Britain and/or France enter the ACW on behalf of the CSA AND the Russians decide to offer something more than moral support to the USA, I can almost see that tempers might calm enough and intertwined alliances be tangled enough for both the North and the South to fight on the same side.