I think it's theoretically possible for a larger Franco-Prussian War (or whatever it might be called in this ATL), but I don't think it could get to the levels of WWI. And each country had its own possible motivations for joining or not.
If Leopold of Hohenzoller-Sigmaringen is crowned King of Spain, France would probably drag them into this mess, as Leopold's candidacy was the casus belli France and the German states went to war over and with Leopold on the Spanish throne, the French would be compelled to kick him off of it. In short, if Leopold becomes King of Spain, then Spain probably sides with Prussia. If the coronation is still averted, Spain probably stays neutral.
Denmark could be compelled into siding with France, because they wanted Schleswig-Holstein back, but unless France clearly beats Prussia and Denmark somehow obtains an actual casus belli, they're staying out of it.
Austria could be compelled to join France. Franz Josef wanted to side with the French in OTL, but was prevented by the Hungarians. If the Hungarians for some reason don't oppose war (which is unlikely) or Franz Josef is compelled to ignore them (which wouldn't be very smart on his part, something which he recognized) Austria probably joins France. If not, then Austria stays out of it, which is what happened.
The Russians could theoretically side with France, but why would they? The didn't have the hindsight to realize the fact that Germany would unify, and Bismarck was always sure to make Prussia/Germany stayed on Russia's good terms, and OTL it wasn't until he left office Russo-German relations soured, so for as long as he stays in office, Germany and Russia would get probably get along (although the minute he leaves office, Germany and Russia will very quickly be on opposing sides). Also, if Austria allies with France, Russia is not going to ally with them. After Austria refused to help Russia in the Crimean War, Austro-Russian relations were TERRIBLE at this point in time. They would rather see a unified Germany than ally with Austria. If Austria allies with France, Russia might actually ally with the Prussians, because they'd rather see a united Germany (which under Bismarck would be friendly to Russia) dominate central Europe than a Habsburg Austria to whom they are actively hostile to do the same. I've also heard of a secret anti-Austrian defensive alliance between Prussia and Russia in effect about this time, but I haven't been able to confirm it, so there's a chance Russia may be treaty-bound to support Prussia. IMO, Russia probably stays neutral or allies with Prussia, but only if Austria allies with France.
Without our 20/20 hindsight, Britain is not getting into this mess. This is the Era of Splendid Isolation, after all. I don't see how Britain would get involved in this. Best casus belli I can think of is somebody stepping on Belgium's toes in some sort of ill-advised proto-Schlieffen Plan, but this is 1870, not 1914. The geopolitics of the time are vastly different, and British neutrality is a near-certainty, IMO.
The Ottomans have their own problems. Unless Russia steps on their toes, they're not getting involved. Most I can see them doing is duking it out with Russia at the time, but only them. No one else is gonna bother with them except maybe Britain and/or France, but it seems very unlikely to me.
Italy might want things France has or wants, but they don't want to go to war with France. They might join the Germans if they fight against Austria, and an earlier invasion of the Papal States might warrant a response from Napoleon III, but I don't think Victor Emmanuel II wants to go there yet. I have an idea where a differing set of circumstances lead Giuseppe Garibaldi to use his wish for battle about this time to try to take Rome again rather than fight for France, which might lead to Italy accidentally provoking France, but it's not a very good idea. The point here being that Italy wants to stay on good terms with France but hates Austria, so they're like Russia: either neutral or potentially allying with the German states if Austria gets involved.
As for the result of this war, the side with Germany still probably wins, as the expansion of the war would likely see more support come in for them than for France. In fact, they may get to annex a large portion of Austria, which would make them much more powerful in the long-term. Spain as an ally of Germany might be cemented in this war, as could theoretically an alliance with Italy. Hungary might come into existence as an independent kingdom if Austria is partitioned (though I'd imagine Bismarck would fight this with tooth and nail, because he would not want Austria as part of Germany).
