One thing I can never understand about this scenario: By this stage, Britain was fervently anti-slavery.
To address this, I need to know: Why is Britain aiding the CSA in the first place? The CSA is a slave-nation, which Britain is not, and Britain is also pressuring the rest of the world to give up this practise.
Judging by the aid/sympathy the CSA got from Britain in OTL, economics, trade, and power politics trumps ideology, at least for the politicians up at the top. There would definitely be unrest among the working classes though, since they tended be among the staunchest abolitionists.
The CSA has cotton, but so does India, which is controlled by Britain.
IIRC, India and Egypt's cotton industries weren't very developed at the time; in fact, it was the ACW that prompted Britain to develop cotton industries in its own colonial holdings instead of remaining dependent on imports.
What becomes of the CSA post war? They cannot sustain an economy based on cotton, because as hindsight tells us, eventually the cotton market will be dominated by Egypt and India. What else has the CSA got? I ask this because I don't know.
Cotton was still a pretty viable cash crop well after the ACW. If the CSA is smart, it can use it's cotton profits to fund industrial development (The ACW did demonstrate to the more prescient folks in the CSA that industry was a necessity). A CSA that can no longer rely on Northern industry is going to have to build up its own industrial base.
The problem comes with the possibility that the people who are smart enough to use cotton profits on development the CSA economy will be outnumbered by the people who just want to pretend King Cotton could last forever. Even then, the CSA economy isn't going to utterly crash until the Boll Weevil hits.
The CSA annexes parts of the USA - we're talking Arizona and Kentucky - but how long can this last? Realistically, isn't the CSA doomed to failure? Will it really be able to sustain an alliance with the British Empire? Or will it eventually gravitate to becoming nothing more than a client of European interests, doomed to be ignored once better opportunities strike its allies?
That's the biggest issue; even if the CSA is smart and spends its cotton money on building factories, the USA already has a huge lead on them, will be growing faster, and has a lot more potential for long-term growth. Isolationism isn't going to be a viable choice, so they'll need an ally, preferably Britain. Keeping Britain as an ally is going to be highly problematic as long as slavery is still around, but arranging any kind of abolition program (even a gradual, compensated one) is going to be very tricky to pull off in the aftermath of the ACW.
I wouldn't say abolition within a few decades of the ACW is impossible, but it would be incredibly difficult to manage, and even then it's likely to be a very slow-moving program that's pretty close to Abolition-In-Name-Only, which is only accepted by the CSA because the alternative would be losing their alliance with Britain and being crushed by the USA.