British intervention might also make a boost for Union morale. Nothing hurts the cause of a secessionist group like foreign intervention; it gives the opening to Lincoln to paint the CSA as a British puppet and further undermine their legitimacy. Considering every CS state except South Carolina contributed at least one volunteer regiment to the Union Army I could see this hurting Richmond's credibility among some sections in Southern society, bolstering the position of Southern Unionists, and giving the North a slight morale bump. That bump will probably turn into a surge when British Canada gets squashed by the Union Army.
Britain jumping in also means the Union Navy can take a different approach: why enforce a blockade when every British ship bound for the CSA, along with the blockade runners, are now fair game on the high seas as military targets? Britain declaring war effectively declares the British merchant marine to be open season for privateering and with the Union pressed by the British Empire and the CSA I could see the Union turning to privateering on a large scale to throw another wrench in the works.
Well, Britain getting involved would probably keep the Irish in line, and butterfly away some of the OTL support for Armistice / Negotiation with the South. There was a great deal of British sympathy for the South during the war, although there was a great deal of Unionist sympathy as well. At least from reading "A World on Fire", it seems that that the country was probably split about 60/40 pro-Union, which is means you'd need probably something even more than the Trent affair to trigger a war.
The U.S. behaving badly in the Trent affair probably makes England a little less willing to enforce strict neutrality visa-vie ship building and monies mid-to-late war, but I doubt it triggers a decleration of war. Well, at least a 50/50 chance it does not. It's not like the British as a whole were chomping at the bits to get involved in the conflict at the time, and many of their politicians were quite aware of the posturing both sides - North and South - were engaged in diplomatically and took a lot from each with more than a few grains of salt.
But say it does go bad, this obviously can't but help the South. On the other hand, I do think a big question is whether or not this happens before or after the Emancipation Proclamation, and how Britain handles that. Unfortunately for the Union, France will probably also come into the War if Britain does, which will be even more problematic. On the other hand, I'm not sure if the other European powers will simply be idle in such a situation either. The Union at least will be looking for allies, at that point.
Most likely England and France are able to secure confederate independence, but probably after a longer war then they bargained for, circa 1869. The situation has changed in Europe, America is devastated. England and France will probably have bigger political problems than OTL, since they basically fought a war for nothing at best and slavery at worst getting a bunch of their people killed. Canada will a confusing place, with probably less of a unified identity, as the Americans who migrated there are either expelled or never truly absorbed....
I doubt it'll play out exactly like Turtledove imagined, but I figure at the very least, you've either created a strong 20th century enemy for Britain in the North, and a lukewarm ally at best in the South, or a pretty much a isolationist North America, and a Canadian Dominion which is likely to be less stable in the future.