I've thought a lot about this question.
The reality is that slavery got an imperial ban due to the Great Reform Act. The middle classes in Britain were heavily anti-slavery, less receptive to the concerns of business owners, and shortly after they got the vote, the ban passed easily. If the American Revolution is averted, it is likely that electoral reform will happen earlier, rather than later as (a) the American colonies have a wider franchise, and that would pressure Britain to follow suit and (b) it's quite likely the Americans would at some point get some sort of representation in parliament, who would be very pro-reform. A ban could well happen in the 1820s. If the French Revolution doesn't happen, it's even earlier: electoral reform could even happen by the 1790s, which would then mean a ban on slavery by the early 1800s.
The question is whether the imperial ban would apply to the Southern colonies, due to any devolved power they have. If the ARW is fought and lost, ultimate parliamentary supremacy is likely to be put on a much stronger footing and it probably will. If the ARW is averted through some deal, parliamentary sovereignty will likely be accepted notionally, but not in practice. This means the initial ban likely wouldn't apply to the southern states.
However, pressure is likely to be toned up in the decades that follow, and eventually something will be passed that even applies to the southern states I think. The South will says its unconstitutional, ignores British liberties etc, but probably won't rebel, as they've got no chance of winning at all if they do. Better to just take the compensation deal, which will likely be on some pretty generous terms.