What about the dominoes that would fall if the war widened beyond the USA/CSA?
The first casualty would be the Monroe Doctrine.
Out of sheer numbers, the British might have had to send its Indian Army across the Pacific. As noted above, this army was a shambles following the Sepoy Mutiny, but plenty of senior officers would argue that a nice little expedition would be the best medicine.
When the Mexicans defaulted on their loans, not just the French but also the British and the Spanish sent warships. Only the French stayed, but in our scenario, the British would not have left as they did, with the expectation that the US would assert the Monroe Doctrine and send the French back home; relieving England of her traditional chore of thwarting France (plus, Victoria personally liked Louis Napoleon).
So the Indian Army would land alongside the French at Monterrey and move north to assist the Confederates. This would take the pressure off Canadian troops on the other border.
Still, Canada is hard-pressed, and the Red River Rebellion burst out and is either successful, splitting Canada, or is quelled even more brutally than it actually was, making Canada a nastier place than it’s become
The Russians could get pulled in (in fact, their fleet did make an extended port call in NYC), looking to reverse their losses from the Crimean War, and seeing India wide open.
Faced with the prospect of World War One in the 1860’s, Prussia steps in to play the “Honest Broker.” This is great, because Bismarck is in charge of Germany as it steps to the forefront of international affairs, instead of Wilhelm II. Prestige instead of war attracts the small German states into its empire, leaving no need for wars with Denmark, Austria, France and then 1914.
(OK, that last one was a stretch, but consider this):
The actual WWI was prolonged by the money and material flowing in from a strong USA, as well as setting the stage for WWII by the meddling of (Virginia-born) Woodrow Wilson. In a divided USA/CSA, Wilson would have had the same authority in Europe as the President of Argentina.
If all that fell into place, then the world would have been a better place today if the British had declared war on the USA in the 1860’s