The Mexican Empire is never formed. Say the US (Lincoln lives?) puts troops on the border quicker, making the ever nervy Napoleon III stick to the punitive expedition and bar sending guns to local conservatives, shelving his plans for the Americas. This allow France to save money, men and international standing plus no Maximilian being shot helps Austro-French relations.
Attention returned to Europe, Napoleon and Franz Joseph team up in 1866, the best possible time for stomping Prussia in terms of it being deemed a legitimate threat, having enough German states arranged against it and not being too powerful vis her continental rivals.
Let the Austrians do better in Bohemia, combined with the French marching through the Rhineland, aiding the likes of Hanover. Bismarck is sent packing and Wilhelm begs for peace.
West-Prussia becomes the Kingdom of Rhineland, perhaps under Maximilian (his connections to France might make him palatable for Paris, Vienna and the Germans)? Hanover gets some nibbles, France buys Luxembourg, and Austria gets Silesia.
All the German states (possibly sans Prussia) form a new loose Confederation which is effectively Rhineland, Hanover and Bavaria under Austrian protection maintaining a divided Germany in dynastic self-interest.
Napoleon III dies mid-1870s, Napoleon IV marries an Austrian princess and liberalises the government to a Prussian level, allowing semi-constitutional rule. Bring in a 'Universal France' immigration policy, which sees European immigration into France rise, combined with some (to our eyes sinister) natalist programmes the metropole has a few million more citizens by the 1920s. France gets a hand in the Rhineland tiller, aiding their domestic industry.
No Germany, France ends up running places like Togo and Cameroon. I imagine Russia cosies up to Prussia after 1866. All in all this leads to France maintaining 'favoured nation' status with Japan which leads to Paris and Tokyo working together in Korea, Formosa and Yunnan, leading to greater French involvement in China.
They avoid a costly war due to Britain being moderately friendly - they see the Austro-French Alliance and Zweikaiserbund as relatively balanced in Europe, while the effective absence of other powers leads the British and French to agree to divide Africa between them (Napoleon IV's Anglophilia helps this, says he becomes mates with Edward VII?). Also fears of Russia see them team up in Turkey and over Japan in Manchuria.
A stronger France also keeps an equal footing in Suez, meaning any invasion in the 1880s is a joint venture, keeping Egypt less of a British colony and at worst a condominium. So no cold war but more a friendly rivalry in colonial affairs, the British confident in their naval supremacy, and the French more concerned with the continent and practising the Jeune Ecole, happy to avoid a ship-for-ship arms race.
However I believe such a set up, based on denying German nationalism and maintaining the Austrian Empire (possibly without the 1867 Compromise so even worse), has a very particular time frame.
Being uber-optimistic, lets say excellent French diplomacy has produced a very loose London-Paris-Vienna-Constantinople-Tokyo entente around Russia and her dwarven barrack friend, Prussia which has become a very peculiar place sitting on her Polish subjects, clamping down on art that isn't deemed masculine and Prussian and basically being rather beefy and dull. I imagine it being caricatured as a sort of Wagnerian North Korea by the Parisian press. Actually that seems exactly the sort of state Wilhelm II was born to rule. I imagine a lot of massive phallic monuments to the Teutons and such, plus probably a tower in Berlin slightly bigger than the Eiffel. A monolithic super-battleship moored off the Pommeranian coast, completely unfit for ocean travel but certainly BIGGEST!
I think it could be fun to write something set in this inferiority complex riddled Prussia, I keep that in mind...
So yeah, France keeps all her plates spinning, maintains a huge empire, first amongst equals in Europe and Prussia is a freak show. Plus population decline has been halted and politics is basically stable. I believe it will all end in a replay of 1848, only this time with socialists thrown in, ending with a republican Grossdeutschland and the Second Empire collapsing at a time when Marxism and fascism could be possible heirs.
But up til 1920-30-ish? A lovely time for France.