While I'd consider an Anglo-American war over the Trent Incident and others (I imagine there would at least need to be a few similar incidents/skirmishes within the same timeframe to push it to war) to be just on the edge of plausibility, the CSA would only be worth supporting because of how it forces the US to direct most of its efforts there, and indeed Britain would be trying to make it as clear as they could (both to their citizens and the international community) that they're only allies of convenience, that they're in the war solely because they themselves were "wronged" by the US, and that they in no way support the CSA other than the distraction they are for the US, or its peculiar institution. Which does not bode well for the prospects of a lasting alliance between the two after the war.
They might be able to sweep the issue under the rug for at least a few years after the war, while the public is distracted by the angry rhetoric between the UK and US, or the Prussian situation. I imagine France is going to suffer an even worse defeat against Prussia than in OTL, after having spent blood and treasure in the American War, and being forced to keep part of their military stationed in Mexico so the still-existent Second Mexican Empire stays propped up. But eventually, the issue of the alliance with a slave power will come to the forefront. It would likely be a galvanizing issue for the political left; imagine workers decrying their government's support for the continuation of slavery and thus the regression of workers' and peoples' rights in general, or citizens boycotting cotton picked with slave labour, which could lead to the election of a government unfriendly to the CSA and in favour of dropping the alliance or demanding the CSA abolish slavery. The Confederate political system made this basically impossible, so dropping the alliance it is.
This is if the CSA's natural tendency towards expansionism (it was always pro-slavery politicians who demanded the US seize more territory down south for the expansion of the plantation system) doesn't kill the alliance first. Once Spain is falling apart in the Third Carlist War, perhaps they could hardly resist the opportunity to pounce on Cuba. Or they invade Haiti because a country founded by a slave revolt is something they can't stand having next door. Or they try to forge an alliance with Brazil, prompting fears among the more progressive wing of British politics that the world's remaining slaveholding nations are standing together to protect and expand their ideology. Either way, Britain would be highly uncomfortable with this and cut them off.
There's also the chance that, as France is forced to pull out of Mexico while busy getting defeated by Prussia, the Confederates take the opportunity to seize its northern states while it falls back into civil war (perhaps as "payment" for offering token support to Maximilian's forces and then spiriting out him and his government when things go south).
By the year 1900, the Civil War-era alliance of convenience would be long dead, repudiated by an increasingly liberal UK that found it politically untenable, and unaffordable to a badly defeated France that now has a powerful enemy next door and can no longer afford military adventures in North America (if they're a democracy like the Third Republic, they would find it just as unpalatable as the UK; it they're authoritarian revanchists because of their worse-than-OTL defeat they may find it morally acceptable, but that won't matter anyway since they can't afford it regardless). The CSA would be diplomatically isolated, left alone with whatever allies it found/coerced in the Americas, and whatever ill-gotten gains it might have opportunistically grabbed.
The UK may have sent out peace feelers to the US, assuring them that their alliance with the CSA is long dead and that they're just as anti-slavery as they are. But even after all these decades, I don't think the US would be in a forgiving mood. They still have a slice cut out of them, the defeat would still be in living memory for some, and of course they would still officially be laying claim to the seceded states. And knowing the advantages they had over the CSA, the blame for how the war went wrong would be laid squarely on the European powers that couldn't mind their own business. Alliances would have been forged by now, likely with the US and the UK in opposite ones, so all Britain can hope for is that the US gets an excuse to go to war with and destroy the CSA before a general war starts in Europe, so they don't have to fight them again. Because they'd be in a worse position than in any previous wars with them if that comes to pass.