IOTL, Britain abolished slavery in the 1830s, thirty years before the United States did, and with considerably less bloodshed as far as I know.
But I have to wonder...if the thirteen colonies, or at least the Southron colonies, remained part of the British Empire, would that push back the abolition of slavery in Britain? Assume that the cotton gin is still invented, or not; it's up to you. Could it even hasten the abolition of slavery?
Sad thing is, I very much doubt there's a decent chance of earlier abolition in Britain's American colonies(well, it really depends on which time frame you use, whether it's 1833 for the OTL Empire or 1865 for OTL's America, though I went with the former here.).
One thing that some may not understand is that part of the reason that the popularity of the abolition of slavery took off, at least early on, was really in opposition to the United States(though, ironically, it was the anti-slavery North that first rebelled, and it was the South that was largely Loyalist!).
Without a successful ARW, there goes one of the biggest motivators for the abolitionists. And another problem that could present itself is, what if London decided to cut their previous losses and cut some deals with Southern planters, even some who may have leaned Patriot? I would imagine that this wouldn't be at all difficult to do and, if that happens, you can kiss any chance of abolition before at least 1835-40 or so good-bye, no matter what kind of bones might or might not be thrown for the Yankees.
I know some may feel differently, but I have no doubts that, if all the "right" chips fall into place, slavery could feasibly survive in some fashion, to least the 1870s, and possibly as late as the dawn of the 20th Century.....and this is in a still unified British North America(perhaps not quite as strangely messed up and dissonant as DoD's *U.S., or as Orwellian as the Confederacy from "C.S.A., the Movie", but could cut it close.).
And should a North-South style split occur, who knows? It might not end until the middle of the 20th century in
that scenario, for all we know.
So, basically, I'm with J.F.P. on this one.