Frankly, I think a World war in the 1860s is unlikely precisely because the politicians of that time by and large tended to be cold-hearted realpolitik engineers. The time for 'drums, guns and glory' had not yet quite come. However, if it were to happen...
Britain might not wish to play, but if Russia comes in, they'll feel they have to. Relations with France were still quite good through much of the decade, but with India still in shock from the Mutiny and the Crimea barely past, I think the idea of a Russian advance on the Dardanelles or the Khyber - or both - wiould be the only thing to move them. (I firmnly beliee the Trent incident was way overblown. Britain would just not have gone to war with another industrial Western nation - an English-speaking one to boot - over such a triviality unless they had intendeed to anyway)
France - I dunno. They are powermongering all over the place, but I don't think they're committed enough anywhere to actually go to full-scale world war. Except maybe in Italy, against the Austrians. hmmmm...
Prussia's main interest are still inside Germany. They might chooswe to have itz out with Austria and its allkies sooner, but their base is still weak and Hanover might get in the way big time. If they finagle a Franco-Prussian alliance that might work. puts Denmark firmly in the Austrian camp - too worried about Schleswig-Holstein to do otherwise. I'm not sure Denmark has any other enemies at the time, but if they end up on the wrong side of Britain, the Great White Queen will hae Greenland for breakfast. Rounds out Canada nicely
What's Austria's current position vis-a-vis Russia? They are opponents in the Balkans, but I don't think it's as bad as all that yet. Might Vienna overcome its suspicions of St Petersburg enough to ally against the French in Italy and the Prussians in Southern Germany? Any such alliance would effectively forestall the Prussians, but it requires the Russians to break with Berlin. Why would they do that? Maybe over some border dispute in Poland? Or over Sweden? hmmmm.
The US will not likely join in any other party - they're firmly engaged in the little thing they have going at home. Ogf course, Europe's powermongers just can't leave off tinkering with something so tempting - a chance to nullify the Monroe doctrine, no less - so there's a possibility someone comes down firmly in the Confederate camp, but I think it unlikely. The French are really the only candidates, what with being engaged in Mexico, and they wouldn't do so without British approval. The Brits are unlikely to agree, especially after 1863. What would they gain? But eve more to the point - the US already had draft riots, anti-black pogroms and a strong anti-War party in a war fought about nothing less than its national integrity. Its finances were depleted, the paper currency inflating daily and the military bleeding to death in huge, costly battles. What would the public reaction have been to Lincoln's 'On to Canada' speech after Appomattox, I wonder? I mean, sure, they'd have taken the place, but at what cost?
As to China, Japan, and points East, I don't think they're in any shape to become belligerents right now.
What about South America? is there any state at the time capable of fighting a war against extnal enemies without risking immediate governmental collapse? Argentina, Chile and Brazil come to mind. I think we have a resident expert on the board, don't we?
So WI we hae a thorough breakdown of Prusso-Russian relations in the course of the Crimean War. Initially, it just opens the Anglo-French fleet more coaling stations in the Baltic. Big Deal. But...
Russia now undertakes to guarantee Denmark's territorial integrity to spite Berlin. This is seen as turning Denmark into a Russian satellite in Stockholm, and Sweden gets nervous. A tentative aölliance with Berlin is in the works.
Prussia and Austria have no reason to gang up on the Danes and instead concentrate on their mutual enmities. Some kind of butterfly makes a rapprochement with Russia look like a good idea to Vienna (I really think it would break down here).
Berlin now feels very lonely in a playground full of bigger kids. Britain has made it very clear it won't commit to anything. That leaves France, where they really like meddling in Italy and Germany. A defensive alliance is concluded in 1860.
As the ACW breaks out, Britain and France decide on initial nonintervention, but France would rather play. Tensions rise, and greater alienation ensues. If this is to go really badly, have it end with Britain backing the CSA. I don't think for a moment they will, but again, it makes the result so much uglier.
If now
Sardinia goes to war in the Veneto
or
Russia goes on the offensive in the Balkans
or
Prussia invades Hanover or Bavaria
or
A revolt breaks out in Schleswig-Holstein
or
or
or
We could have Austria at Prussia's throat, with France and its Italian allies taking it in the back (or vice versa). Russia makes a threat, gets its bluff called, and has to commit. Either Russian troops get too free with Turkish territory in the Balkans or the Black Sea fleet insists on going through the Dardanelles, at any rate they don't need much of an excuse for a war with Turkey. That'll bring in Britain, especially if the Russians are dumb enough to drop hints about Cossacks on the Khyber.
Could we get Spain in through being worried about French power projection in Mexico? It still has Caribbean possessions. Would be military suicide, though.
Depending on how things stand in the Americas, we might have them involved. The US might get ideas about Canada, but I doubt it, especially as that would effectively mean plumping FOR Austria and Russia. Maybe, though, France is dumb enough to declkare war over Maximnilian unilaterally. Make Martinique and Guadeloupe US holiday spots? hmmm - then the US Navy goes on to seize St Pierre and Miquelon which is stupidly provocative. That really could lead to a war with Britain.
Either way, the Austro-Russians lose. Depending on how long and nasty the war gets, they might suffer quite severely. I can see a worst-case scenario where Prussia gets control of most of Germany all the way to Bavaria (not Austria proper, but quite possibly Bohemia as well), with a guarranteed frontier along the Rhine, the Veneto, Lombardy and Alto Adige go to Sardinia, Sweden gets bits of Finland, maybe even the Baltic states (though that would be HARSH), Denmark loses Schleswig-Holstein to Prussia's 'German League' and Greenland to Britain (if London wants it). Alaska may have to be ceded to Britain and some Polish territory to Prussia. There will be strong guarantees for Ottoman integrity in the Balkans (like they haven't been given before) and indemnities to be paid. France gets to run Germany and Italy almost unopposed (until they figure out Berlin and Turin aren't playing along like good kids all the time).
If there's a war between the US and Britain in America, I think the US would take Canada, and the Royal Navy would have the US Navy for a snack. Then there'll have to be a diplomatic solution because the loss of Canada humiliates Westminster enough that they need it back, but can't take it by military means, and the British blockade (and possibly operations in California - without a transcontinental rail link, the logistical advantage in the Pacific is on the Royal Navy's side) hurt the US enough economically to make them want to compromise. Especially if the British throw in something symbolic, like The Shelling of Washington. Also, I'm sure the Canadians won't make happy subjects,and that means the US need to hold down the south AND Canada - ouchie.
However, this development is still a tale of stupidity quite out of character for mid-19th century diplomats.
