I mean, its a really interesting idea. So Napoleonic France wins in Mexico, and in turn creates closer ties to Austria. Due to this Bismark fails to secure an assurance of neutrality from France, but due to his need to unite North Germany (and trying to contain a liberal legislature) war does break out, and France ultimately declares war on Prussia. Prussia loses, and loses some of its territory to France, a bit more to Austria, and a few more German states are formed from that territory. With Austria now in a dominant position across Germany Prussia, still a power on the level of Austria military speaking is forced to set aside goals of German unification. The lost war completely discredits Bismark, and the King is forced to make concessions to the legislature.
Meanwhile in the United States, the Democratic president following Lincoln was a dismal failure. While the Republicans lost the war they also pointed fingers at the Democrats, accusing them of sabotage, and in 1868 the Republicans spring back, running Ulysses S. Grant for president. Grant was one of the great successes of the war, driving down the Mississippi, forcing surrender after surrender (and okay there were a couple setbacks, but it was in the east the war was lost); and is elected. OTL of course the Grant administration was plagued by corruption, but with the need for reform clear this is significantly lessened. Furthermore, Grant and the Republican Congress decide that in order to prevent another humiliation at the hands of the Confederates and their foreign allies that the United States needs a solid, professional army. They turn to Prussia, which needs cash after the loss to Austria, and is still renowned for military prowess. Congress approves the purchase of designs for Prussian rifles and cannon, and sets up supplies for an expanded US Army, say 75-100,000 men with additional reserves formed from the veterans who were discharged from the Civil War. With Prussian officers providing expertise the US Army professionalizes by 1880, when the CSA decides to approach Mexico and basically force them to sell Sonora and Chihuahua. US President Thaddeus Stevens, facing possible defeat in the November election decides to drum up support by declaring that the United States will oppose any attempt by the CSA to take the two states.
Meanwhile the Confederacy, riding high on its victory returns to the same system of state militias and continues to believe itself invincible against the damnyankees. They ignore it and move in when Maximillian reluctantly agrees, and the United States declares war.
What follows is one of the most one-sided modern wars ever. Britain, its government in a quandary about how to proceed half-heartedly backs the CSA, while France is too distracted elsewhere to get involved. The American Army, and a larger Navy descend upon the Confederacy with a disciplined, well-trained, and well-supplied force. As the Confederate state militias are rallying Kentucky's capital falls, and the Virginia State militia is scattered in one of the only major battles just north of Richmond, having failed to even slow the professional Union men. In ten weeks Richmond falls and the Confederacy is facing utter defeat.
Basically, the equivalent of the Six Weeks War or the Franco-Prussian War by comparison.
