I'm been doing some R&D for a TL of mine and it got me thinking about something. In 1866 the Prussians crushed the Austrians then five years later the French. What if after these two wars the Austrians and the French form an alliance to count act the growing weight of Germany. What effects would this have on Europe if such an alliance would come to pass?
I think this actually was attempted before 1866 as a counterweight to British and Prussian might. Hapsburg Mexico was to be their counterweight to the United States. Though I'm not sure this alliance would preserve Mexico--if France enters the war against Prussia in 1866, then French troops will be withdrawn, and Max's government will fall. And troops won't be sent back to Mexico afterwards, since the US by that point will be able to enforce the Monroe Doctrine.
If it were to come to pass, then joint Franco-Austrian action in 1866 might be able to stop Prussia from completing the unification of Germany. Otherwise, Austria will fall into the Prussian/German sphere of influence.
So, assuming victory against Prussia in 1866:
Prussia is still out to unite the Germans under its rule. It will be opposed to Austria, and thus lean toward alliance with Russia (which, through Serbia, is also an enemy of Austria). This means that the Ottoman Empire will join the Franco-Austrian pact. Mexico's fate is up in the air. But it might be undesirable to continue to support Mexico, as that would antagonize the US and drive it into support for Prussia.
Bulgaria: a natural Russian ally, and Russia will still want a Mediterranean port. I can't see a Congress of Berlin taking place in this world, so maybe the Treaty of San Stefano holds.
But then that makes Russia well-placed to carve up the Turkish Empire into satellites, which would antagonize the British. So maybe the British will end up backing the Franco-Austrian-Turkish alliance against Russia-Bulgaria-Prussia-Serbia. But there'd be no naval arms race, so Britain would not have particular opposition to Prussia--you could end up with a situation similar to the USSR in WWII, at war with Germany but not Japan until the last days. Britain would be opposed to Russia and Bulgaria, but not necessarily Prussia.
So what I'm seeing is France-Austria-Ottoman Empire against Prussia-Russia-Serbia-Bulgaria. Britain could go either way, but I suspect they will be primarily anti-Russian. The US could go either way (depending on Mexico, and Britain's stance wrt. Prussia).
EDIT: Ah, but that assumes Franco-Austrian neutrality in the Russo-Turkish Wars. Which, I think, is likely, but not necessarily the case...