I think this question is tightly linked with the continued survival of the CSA itself. And I personally think it is likely that the CSA will collapse before it finds the will to abolish slavery on its own.
First of all the CSA is guaranteed to be embroiled in disputes with the USA. One key issue will be runaway slaves. Before the war the fugitive slave act mandated that any slave had to be returned to its owner, and anyone who assisted him faced sever punishment. The northern states hated this law and people accused of violating it were acquitted by juries making its enforcement almost impossible. Thus after the war there is no way the USA is going to return escaped slaves, which will be a constant cause of friction. Secondly, there will be the question of settling the west. Bleeding Kansas was not only a prelude of the civil war but also the second major cause of it. The peace treaty of the civil war will doubtlessly include some kind of settlement for the western territories, but American settlers were not know for respecting such agreements. In Arizona and New Mexico there will be fights between settlers arriving from the USA in the west and the CSA in the east, in Oklahoma and Colorado there will be clashes between settlers from Kansas and those from Texas.
Thus I would argue that future conflict and wars between the USA and CSA are inevitable. Now as the industrialisation progresses the balance of power is going to shift further in favour USA, and the CSA can only hope to win a rematch with outside help. In fact, the general consensus of this forum seems to be that the only way for the CSA to win in the first place is due to help from France and/or Britain. Now, Napoleon III might be willing to prop up the CSA indefinitely to safeguard his Mexican adventures, but the German unification is going to put and end to that [1], just like it ended his support for the papal state IOTL.
The UK on the other hand lacks any strong incentive to keep supporting the CSA. On one hand the population hated slavery (and as has been noted the UK had been curtailing slavery for about half a century), on the other hand the USA was a major trading partner, which is not going to change because of one war. Furthermore the CSA cannot offer the UK any support in areas where it matters (India and China), so the alliance offers little gain for Britain. And historically Britain had absolutely no problem to end alliances under these circumstances.
So I'd say that slavery lasts as long as the CSA does and ends sometimes in the 1890s after a particularly bloody massacre of between settlers from the CSA and USA in Colorado triggers a second war.
[1] Otto von Bismarck was appointed as chancellor of Prussia in 1862, the Second Schleswig War took place in the fall of 1864, and the rivalry between Prussia and Austria in the German Confederation had been ongoing for a decade. Thus, I'd say that it had aquired sufficient inertia to be stopped by a few butterflies.