Portugal either stays out and profits through trade or engages in token involvment due to the long-standing friendship and alliance with the UK.
Spain really isn't going to be happy. It has nothing to gain either way but CSA independence...the CSA has made far too many noises about grabbing Cuba.
The Ottoman Empire's involvment would be disastrous for any allies. Their defensive capacity is uncertain, their ability to attack is extremely doubtful, and their direct involvment all but guarantees substantial Anglo-French forces needed to shore them up.
Polish uprisings last even less time than OTL if Prussia is working with Russia.
In 1866 Prussia flattened Austria, so it is very doubtful that they won't be able to hold their own four years earlier, except that now Russia is on Prussia's side. A scratch Prussian force holds the French as long as possible, while Russia and Prussia blitz Austria and maul the Ottomans. Then Prussian armaments and Russian numbers move west...if Prussia turns on Austria really badly we might see Hungary free and the Balkans(of not the slightest interest to Bismark) set up as several smaller states which the Russians will like, with new royal families related to the Hohenzollerns. Of course, if this becomes a long-term Prusso-Russo partnership...
Italy stays neutral. France does what it did in 1870, abandoning the Pope, and also persuades Austria to relinquish Venetia, as Napoleon III offered in 1866. With substantial gains and no fighting Italy has the last laugh.
Mexico may avoid war as well, as France needs every man at home.
Note that following the Prusso-Austrian war of 1866 France credited the Prussian 'needleguns' and raced to replace their own rifles. Therefore France was not so equipped in 1862. Prussia, having been so equipped in 1866 is very likely to be at least partially equipped with such rifles in 1862 so any French advance will be bloody against even a moderate Prussian defense. Then the hordes from the east arrive with the bulk of Prussia's armies. Not good for Paris.
Note that in 1866 a scratch corps of Prussians was able to mop up ALL of the lesser German states on Austria's side.
Note also the effect on French moral and attitude towards Napoleon III as they abandon the Pope and the Mexican adventure to defend themselves. Alternatively, the attitude if trouble begins while 50K or more are off in the Vatican and Vera Cruz. Poor Napoleon Three, heads Bismark wins, tails...
The British position is interesting as the tiny British Army is now dangerously liable to be hopelessly overstretched. Consider the needs:
1) There has to be a contingent with the CSA of some kind of size.
2) There must be a force large enough to help the Ottomans at least as far as holding Istanbul. If Russia reaches the Bosporus and Dardanelles, it really is all over and the war is a certified catastrophe for London.
3) They must send a substantial force to Canada and soon. If the Union throws a strong army(50,000?) and seizes key points on the St Lawrence then the Union has at least a year in which the British are stuck in a bloody campaign attacking fortified positions while the Union can do as it wishes with the bulk of Canada.
NOTE: This presumes a war to be resolved in terms of the 'state of possession'. There is an exceptionally ugly possibility here which I shall get into later.
4) Obviously if France falls, the UK faces an European continent entirely hostile or disinterested so troops are needed there also.
Now, the British Army during the war against the Zulus in 1879 could field less than 130,000 men and many were needed where they were. This leads to the probability of the trained veterans being thrown away early. An early defeat for the British could be very dangerous, and British officers are smart enough to realize that. Nonetheless, in several locations aggressive action may be a necessity so if the British send too many to Canada and suddenly the Ottomans crumble...
To make a long story short, British army involvment is going to involve almost every unit being rushed somewhere followed by six months or more putting a larger force together during which reserves will be extremely thin.
One important note: This war is liable to resemble the long series of French and English wars in the 17th and 18th centuries, in which the results on the ground often clashed sharply with the terms of the treaty. In other words, if Great Britain is doing very poorly in Euopre any gains in North America are liable to be sad memoriesof what might have been.
One must seriously question whether, if France also falls, Great Britain is ready to fight a much bloodier version of the Napoleonic Wars against much stronger enemies, and all of it started for a slave society.
I mentioned the state of possession, which basically means that if you hold something when peace talks convene, you keep it. There is liable to be an exception to this rule. Contrary to the beliefs of some, the US was not drooling at the thought of overrunning Canada in 1862 and while the idea of CSA independence in exchange for the western half of Canada might have been seen as eminently reasonable in Paris, Berlin, or St Petersburg(if not so pleasant in London), it is not so certain that the US would agree.
Of course, if the US is doing well on both fronts the British may cede much of Canada and retire from the war in disgust while the ACW then resumes its normal schedule.
Although a CSA certain that foreign aid will not come might negotiate a settlement.
However, if the US is doing well in Canada but barely holding its own vis a vis the CSA/British forces, perhaps on the defensive, what happens if the US explains that they don't want Canada, but very much want the CSA and will gladly relinquish any Canadian soil held in return for a very...perfidious sort of favor...heh heh heh...