The Franco-Prussian War is not only unlikely to go the same way as in OTL, it is unlikely to happen at all. In OTL, in an environment where Prussia had already done most of the work defeating Austria in the Austro-Prussian War (a shock defeat where most people thought Austria would emerge triumphant and yet in fact it was a stunningly swift Prussian victory) Napoleon III issued a threat to Prussia seeking territorial concessions in return for staying out of the war. The Mainz threat was an unbelievably, incredibly stupid move (giving the lie to the popular presumption of modelling historical figures as game-theory-esque perfect rational actors; it was just a really dumb thing to do) because not only was it unenforceable given the current military situation, not only was the French army in the process of re-equipping its infantry weapons with the more advanced chassepot and thus it would have been better-timed to do it later, not only was it stupid because of the Prussian army's recently displayed mastery of mobilisation and logistics thanks to the reforms of Moltke, Roon and Wilhelm I, it was a tremendous foreign-policy facepalm moment because it drove the South German states, terrified of French aggression, into Bismarck's waiting arms. (That alignment of those states with the Prussian-dominated North German Confederation was far from inevitable, contrary to what nationalist historiography may lead one to believe. When the Franco-Prussian War began a lot of Bavarian peasants burnt their crops—a very serious thing to do, for a peasant—so that the enemy couldn't get them—the enemy being the Prussians, not the French. At the time the Emperor of Austria quite seriously contemplated entering the war on the French side if the French were advancing into Prussian-held and Prussian-allied lands and there was a sufficiently strong anti-Prussian reaction from the people there, and the Emperor and Autocrat of All the Russias took this prospect seriously enough that he threatened Austria with Russian mobilisation if Austria mobilised against Prussia.) If Napoleon III does not issue that threat, this diplomatic alignment of the South German states doesn't happen.
If Napoleon III is too tied up in the Americas, in addition to his other commitments in Europe, he probably doesn't feel that his position is strong enough to issue the Mainz threat in the first place, so there is no moment at which Prussia is in the supremely advantageous position which it enjoyed in 1870 in OTL: blatant French expansionism at the expense of the smaller German states had been shown to be a major possibility and Prussia had shown itself willing to stand up strongly against French expansionism just at the time that Austria had just been weakened and discredited as a force to protect the smaller German states and Prussian power had been recently demonstrated. The scenario sounds like it was tailor-made to achieve Prussia's goal of uniting itself with the smaller German states in a federal German polity that would exclude Austria and thus be dominated by Prussia, though I'd argue that it arose more due to Napoleon III's stupidity than to Bismarck's genius. That perfect scenario, in addition to the (fully justified) conviction of the Prussian government and army that this was a moment where they held a major military advantage, made Bismarck decide that now was the moment to aim for this ambitious Prussian goal, so he deliberately escalated the Spanish succession crisis into the Franco-Prussian War and the rest is history. In fact the Kingdom of Prussia enjoyed so many military advantages over France at this point, as much due to French incompetence (in particular the spectacularly bad mauvais numéro system of how to get soldiers and a grasp of logistics so poor that not long ago France sent an army into Italy and only then, after hearing about that army having to beg and work from the locals, thought to send some supplies!) as Prussian brilliance (an extremely swift mobilisation well-orchestrated by a highly organised general staff and enabled by an excellent railway network, thanks to Moltke and Roon, a far superior system of conscription which was later adopted by practically everyone in recognition of this, and the excellent artillery of Kruppstahl), that even without the supremely excellent diplomatic situation of alignment with the South German states the North German Confederation would probably have achieved a decisive victory over Napoleon III's France in such a war anyway, but the extent of Prussian victory in OTL took pretty much everyone by surprise and in OTL Bismarck was too cautious to risk such a scenario without this supremely excellent diplomatic situation.
This sort of concern is why TLs like "The Confederacy becomes independent and then it and the USA take part in the First World War" are so silly. If the Confederacy exists, even several of the major protagonists of the First World War (the German Empire, the French Third Republic) might well not exist (instead there'd be the North German Confederation plus independent Bavaria, Württemberg et cetera, and the French Second Empire), let alone the war itself.
It is of course not inevitable that the Second Empire would rule France indefinitely, as there was opposition to Napoleon III, but I think it likely that without such a disastrous defeat as in OTL it would have survived. Napoleon III was engaging in a degree of democratisation which reduced pressure on the autocratic elements of his regime, and he had conservative Catholics—who were probably the greatest threat to him if things went pear-shaped, judging by the fact that when France did have democratic elections in OTL, even after the dramatic failure and humiliation of a conservative royalist regime in the war, there was a majority of conservative royalists, and the only reason a monarchy wasn't restored is that they couldn't agree on which one due to the flag dispute and the Comte de Chambord's stubbornness—mostly on-side thanks to the Roman question. The widely vastly exaggerated chances of the Paris Commune aside, the Second Empire was likely to survive, absent its catastrophic defeat in war and personal humiliation of its emperor at Sedan.
The absence of the Franco-Prussian War also has massive butterflies on all the militaries of Europe, as you take away the lesson that Moltke and Roon were basically right and everyone needs the general staff detailed planning of logistics that they supplied. Before that realisation, things like supplies being sent to armies by rail and then unable to be transferred out and waiting uselessly for the whole war were commonplace. It is almost impossible to overstate the military importance of such a PoD. I know very little of military history and I can't speak of such things in proper detail, but there are those on this board who can be helpful there.