It all depends on how the war drums are beating. There’s about two and half months between Prussia’s initial mobilization and war actually stating in 1866. If it looks like other countries are getting involved in this time period, Bismarck probably backs down and there is no war. He just waits for the next crisis to try and exploit.
The rapid speed of Prussia IOTL and decisive defeat of Austria in just a month shocked most of Europe, which is why nobody else got involved. France, UK, and Russia all thought it be a longer and more protracted war so they figured they would sit it out and only get involved if things looked dicey. They never had a chance. So the most likely scenario to get a wider European War here is Prussia does worse in the initial stages and events spiral from there.
Russia is slow to mobilize and France already has over 100,000 men committed to Mexico and Algeria, so they’d need time to pull them back to really get involved. Honestly, I’m not sure how much the UK could do at this point. How many men do they even have to commit? The British Army was a minor partner in the Crimean War, for example. Denmark may jump back in and allow Britain to stage there in the hope of getting Holstein back and box Russia in the Baltic, but maybe not since they were friendly too to Russia. But if Russia is backing Prussia that might change. It’s complicated!
I’m of the opinion that this scenario ends in a negotiated peace in less than two years, probably one year, like most European Wars of OTL prior to WWI. They all have institutional scars from the Napoleonic Wars still and the industrial capacity required for full mass mobilization war is not flushed out yet. There are going to be winners and losers on each side.
A lesser Prussian victory versus OTL seems likely, probably a smaller North German Union with less annexations. They don’t gain Alsace-Lorraine but there might be a future war with France still.
Italy is probably going to do poorly. Austria did well IOTL Austro-Prussian War against them and we saw France had a successful rapid mobilization into northern Italy IOTL Franco-Austrian War. If I was France, my plan would be to plan a defensive mobilization against the Rhineland but go on the offensive against Italy. March on Florence and get a ceasefire as soon as possible, secure Rome then move the troops north. The peace probably allows France to secure Rome’s independence for the Pope and withdraw their troops after the war. Napoleon wanted this more than any territory in Italy. Italy may have to give up on Venetia for now which is going to cause nationalistic problems at home and likely down the line for the Austrians.
I’m struggling to see Russia wanting Austria’s dismantled at this point, but it could go either way. In the 1860s. The January Uprising is still fresh and Alexander II seemed opposed to awakening national sentiments. It depends on which way the war develops.
Russia just had major uprisings and basically martial law is in effect across most of old Poland-Lithuania. Do they really want to add more non-Russians now? If Russian engagement is minimal, slow to mobilize and on the edge of the real fight, may be satisfied with simply getting the Treaty of Paris overturned and remilitarizing the Black Sea.
If it turns into a slog, they are already pissed at the Austrians and could be out for blood. But this might be limited to annexing Galicia (partial or whole) and/or Krakow. The Concert of Europe was real and most every diplomat of the time was bought into wanting to keep the ‘balance’. In their minds, Austria played in important part in that system, otherwise Russia could March straight to the Adriatic! If it looks like Prussia is gaining in Germany, the negotiations requiring some tranfers of Polish territory isn’t out of the question.
I disagree that the Ottomans are certain to sit this out. If Russia looks too successful against Austria, they are definitely motivated to jump in as any gain for Russia threatens their position.