It surprises me how many people would think the British and French would come to the aid of the CSA despite having no real reason to. Cotton isn't much of an issue and I doubt they would want to associate themselves with the CSA anyway.
Meanwhile, it's time for the Southern slave supporters and traitors to be reminded about liberty...
What we're really talking about here is not so much the risk of Anglo-French military intervention, as to try to halt the war through mainly diplomatic means.Dude, the British wouldn't be touching that with a ten foot pole at this point... the British public was overwhelmingly pro-union OTL, and with an even more fervently anti-slavery President in charge than OTL, that would be enough to get the House of Commons to stand firm against any aid to the Confederacy.
And the fact is that - even setting aside the Trent Affair - that actually was not improbable in *our* timeline, especially in the late summer and fall of 1862:
And mind you, the Palmerston government was Liberal, not Tory!McClellan’s failure on the Peninsula, Pope’s inglorious campaign resulting in his crushing defeat at the second battle of Bull Run, during the summer of 1862, had a profound influence on the governors of England. The correspondence between [British Prime Minister Henry John Temple, 3rd Viscount] Palmerston and [British Foreign Secretary Lord John Russell indicates that they were about ready to propose to the Cabinet that England should take the initiative and ask France, Russia and the other powers to join her in some intervention in the struggle in America. The Federals “got a very complete smashing,” wrote the Prime Minister on September 14; and if Washington or Baltimore “fall into the hands of the Confederates,” as “seems not altogether unlikely,” should not England and France “address the contending parties and recommend an arrangement upon the basis of separation?” Russell replied: “I agree with you that the time has come for offering mediation to the United States Government with a view to the recognition of the Confederates. I agree further, that in case of failure, we ought ourselves to recognize the Southern States as an independent State.” He suggested, moreover, a meeting of the Cabinet, and if a decision were arrived at, to propose, first, the intervention to France and “then on the part of England and France to Russia and the other powers.” When Palmerston replied to this letter, he was watching the Antietam campaign, and thought that if the Federals should sustain “a great defeat” it would be well to proceed with the project of mediation; but if “they should have the best of it we may wait awhile and see what may follow.”
Gladstone, the Chancellor of the Exchequer and the third member of the Cabinet in importance, was well aware of Palmerston’s and Russell’s attitude and, feeling certain that such would develop into the policy of the government, anticipated this probable event in a speech at Newcastle on October 7, wherein he expressed positively the view of the Prime Minister and Foreign Secretary as well as that of most of the aristocracy and higher middle class. “There is no doubt,” he declared, “that Jefferson Davis and other leaders of the South have made an army; they are making, it appears, a navy; and they have made what is more than either—they have made a nation. We may anticipate with certainty the success of the Southern States so far as their separation from the North is concerne, Hd.”
(James Ford Rhodes, History of the Civil War, 1861–1865, Ch. VII)
It's true that there was little appetite in Britain, especially in the wake of the Crimean War, for a major foreign military intervention, especially against a major industrialized power (one which by 1862 had the capability to overrun most of settled Canada, as @Gaius Julius Magnus point out). It's also true that slavery made the CSA in particular a less than appetizing ally, no matter how many Lancashiremen the cutoff of Southern cotton put out of work.
But it was also the case that even among British liberals, there was much about "Yankee democracy" that was deeply off-putting - its crassness and mob-ishness, to say nothing of the growing threat the United States posed to British trade, industry, and its interests in the Americas and East Asia. There might not be broad enthusiasm for the South, but there was also plenty of interest in seeing the United States, such as it remained, taken down a peg or two.
Palmerston's idea was to offer a mediation once it seemed clear that the Union could not prevail militarily. If the Lincoln Administration turned it down flat, the threat would be made that Britain and France would formally recognize the CSA - with the possibility of trade and aid, and plenty of other European powers soon following Britain's and France's lead. Politically, even just that would be a deep blow to the Union war effort, because it would grant a much greater legitimacy to the Confederate cause, even setting aside whatever tangible assistance it might gain out of the situation: the Northern peace movement would be greatly emboldened. After that, the next risk of escalation would be that the British and French navies might try to break the Union blockade (something certainly within their capability, though it would become a steadily more expensive proposition with each season that the Union Navy built up its strength) - another lever to try to force Lincoln to the table, and put an end to the war.
But let us distinguish between late summer 1862 OTL and spring 1861 in Red's timeline. It's a less "mature" situation in America, and Palmerston's government has had far less time to come to grips with the crisis; and the CSA has not had a couple full campaigning seasons to prove its mettle or give flesh to Gladstone's claim that "the leaders of the South have made an army; they are making, it appears, a navy; and they have made what is more than either—they have made a nation." But if weeks and then months go by with Confederates in control of Washington and Baltimore, they might reach a decision that it is time to offer mediation a lot sooner. The North looks pretty hapless; the rebels look potent enough to occupy even the federal capital, with the fourth largest city (Baltimore) in the old U.S. into the bargain.
And it really does depend on what Britain does. Napoleon III might have had a greater incentive to see the Union whipped, but he also made it pretty clear throughout the war that he was unwilling to move forward without the British doing so, too.