I all honesty I think that slavery in the Confederacy would continue well into the 20th century.
It would be scarse in Virginia and its Upper-Southern neighbors by the early 1900's as they were in the process of downsizing Slavery as an institution in those states in OTL when the ACW began. I would guess that by 1930 Slavery would be almost extinct in Virginia and Arkansas and it would be close to that in North Carolina, Tennessee might hold on a bit longer but would be close behind the other Upper-Southern states in abolition.
The Deep-South on the other hand has more need for slavery on the whole and it would be unlikely that they would even begin to think about abolition before 1900. It would be most likely that one of the Deep-South states, most likely Flodrida, would begin to downsize the institution around the same time Virginia would be close to abolishing it in the 1930's. South Carolina, Alabama, Georgia, Mississippi, Louisiana and Texas would hold on to the institution longer simply becuase they would have more need of it.
That being said I think that Slavery on the whole would be mostly gone from the Confederacy by the 1960's and certainly wouldn't be around in the 2000's in any large-scale form. Slavery in the Confederacy might evole into a more domestic version of it rather than the thousands of slaves working on the plantantions. In that kind of form there would simply be no need for any one slaver holder to own more than 20 odd slaves, at most, for menial chores around the home.
Of course that is only in the form of a continuing slave-holder state rather tha a free one but it is the kind of evolution of the institution of Slavery that I think would likely occur in a victorious Confederacy scenario.