Well, I guess it is theoretically possible that the independent CSA (especially a deep-South-only CSA, which would contain all the craziest fire-eaters with much fewer moderates to balance out their influence) would rather commit economic suicide than abolish slavery even post-Brazil and Cuba, but I suspect that under some pain the non-slaveowning-population (and probably some slaveowners even) would sooner decide that having a decent economy & not being an international pariah (again, the CSA would be the only country in the Western civilizational sphere to still have legal slavery if they still have it post-Brazilian abolition) is well worth at least nominally abolishing their Peculiar Institution.
And that's assuming they don't commit
national suicide in a more direct fashion. Like by restoring the slave trade (
which some fire-eaters really were braindead enough to propose IRL) and then freaking out when the Royal Navy starts seizing slaver ships as they often did, whether at sea or by directly sailing into Confederate ports to do so (
which they were doing to Brazil by about 1850), perhaps to the point of doing what Brazil was too smart to do and opening fire on the Bongs for violating their national sovereignty. I cannot imagine such a conflict would go terribly well for the CSA, especially not for a Deep-South-only CSA most likely to reach this precipice in the first place and then still commit such a blunder (as it'd be more of a backward agrarian shithole, Virginia in particular contributed much of their industrial capacity historically - IIRC 1/2 of the CS military's cannons were produced at Tredegar, the Gallego flour mill in Richmond was the largest of its kind in the CSA and the world before its destruction, etc.) and if it really comes to that, Britain can comfortably demand abolition as part of its peace terms in victory.
I'd argue such extreme measures were not needed in the cases of Spain and Brazil, which were more compliant with British efforts to undercut the slave trade and in general had less of an extreme ideological attachment to slavery & the racial politics further undergirding it. Again, for example, while Brazil did try to skirt their various anti-slavery accords with Britain at first, they gave up once the Royal Navy started sailing into their harbors and seizing slaver ships under the (silent) guns of their forts.
The CSA meanwhile would have to
bring back the slave trade decades after the USA agreed to abolish it and Britain has made it abundantly clear that it'll crush the trade wherever it can, and admittedly they did have a not-insignificant cadre of politicians/intellectuals who were insane enough to advocate such a policy. Not some nobodies either, but people like
William Yancey who was an architect of the Breckinridge candidacy in 1860 and basically one of the CSA's more extreme 'new founding fathers', and whose legacy would be unchallenged in a 'CSA secedes without a shot fired' scenario. I do not believe it is inconceivable that Britain would explore options they never felt the need to (as said previously, for example, slavery was already visibly waning and Brazilian slaveowners were manumitting their slaves at a higher rate for some time before the Lei Aurea's passage due to a combination of British pressure & other factors) in the other cases.
And of course, as I said above, we also have to consider the possibility that the Confederacy might actually be dumb/crazy enough to go straight to war with Britain in response to the RN doing their usual anti-slavery business. Which is not something Spain or Brazil ever dared to do either. I think
@S'task has made a good case that Virginia at least might stay out of this Confederacy without Lincoln moving to suppress the initial secessions, which would not only terribly hurt its industrial capacity but also throw the internal political balance in favor of the wackiest fire-eaters (one of the more prominent guys to challenge Yancey on the slave trade issue was Virginian
Roger Pryor, for instance, and Pryor was already an ardent secessionist and slaver himself, he just wasn't as demented as Yancey).
In that case I think the odds of the CSA blundering into a full-on war with the UK over the slave trade spikes upward (a CSA psychotic enough to bring the slave trade back is one that I think will be psychotic enough to open fire when the Royal Navy inevitably starts raiding their ports for slave traders as was done to Brazil, realistic assessments of their fighting strength vs. Britain's be damned), and I really don't think that's a conflict they can hope to win (especially not in a Virginia-less condition, VA was really critical to their already limited industrial capacity in the real ACW). If it really comes to that - well there's no historical precedent for the Brits directly abolishing slavery at the point of their (pre-)dreadnoughts in a western country either, but this would be a golden opportunity to set such precedent.
As far as cotton consumption goes, Britain I know for sure was able to comfortably make up for the drop in Southern cotton with Indian & Egyptian replacements when the CSA tried to blackmail them over 'King Cotton' historically. I'm not terribly sure any foreign power would be able/willing to help the CSA out against Britain either. If the course of history outside of the Americas still looks anything like it did IRL, the French are going to want British help as a counterweight against Prussia/Germany; they were willing to eat a humiliating L on Fashoda to appease Britain for the sake of the anti-German diplomatic strategy, why would they stick their necks out for the CSA, and over slavery of all things? Russia was generally pro-Union, had decent relations with the US (even outside of the Civil War context) before the October Revolution and all the ensuing problems to my knowledge, and can't really project power to help the CSA in any meaningful way to begin with; the Great Game between them & Britain was also starting to wind down around the turn of the century.
Germany's diplomacy in the Americas from this time period - the only significant actions they took that I can remember is that 1)
they worked with Britain to blockade Venezuela over some debts, 2)
that time Kaiser Wilhelm went full supervillain and dreamed up an invasion of New York for reasons that didn't make a whole lot of sense (fortunately for Germany, the plans remained theoretical and were ultimately completely shelved) and 3)
German Haitians controlling much of that country's economy by 1910. I guess they'd have motive to want to take Britain down a peg, but I really can't imagine them being able to do a whole lot in support of the Confederacy to do it, especially if those nutty fire-eaters have gone and started a direct war with Britain.