In the real world, Parliament passed an act to (gradually) abolish slavery in most of the British Empire in 1833. Of course, that might have triggered a revolution in some of the Empire’s North American dominions right there. On the other hand, real-world Virginia had a debate about abolishing slavery in 1832 (in the aftermath of Nat Turner’s rebellion). Perhaps the wishes of South Carolina and the other slave [del]states[/del] colonies would not have loomed as large in an Alternate History British Empire. The AHBE might also have been more centralized than the real-world United States, with Parliament able to ignore the wishes of its colonial dominions in a way the U.S. Congress can’t generally do to the states–though that might depend on exactly how the British Empire in the Thirteen Colonies survived.
I think that’s a bit of a different question. If there had been no American Revolution (or the American Revolution had failed), it seems very unlikely that a British colonial subject named Franklin Delano Roosevelt would have been born “on schedule” in Her Majesty’s Province of New York in January 1882 (and still less so that he would go on as Prime Minister of the United Empire Parliament to lead the Empire to victory in a great World War against an alliance of Germany and Japan)–barring some (to me really unlikely) ironclad force of destiny or fate, there would simply be too many random and not so random changes in the timeline for such a person to have ever existed, let alone led some kind of parallel life with the real FDR.
To put it more personally, if the Axis had won World War II, I would not exist. Just to look at one factor out of many, one of my grandfathers was a bomber navigator in the U.S. Army Air Force in the European theater. Rather than returning home to father my father, in a “Hitler Wins the War” timeline he would have been shot down and killed in the Battle of Buffalo, or died of typhus in a German P.O.W. camp in 1945, or simply been locked up in a German Stalag until the Armistice of 1947 (my father was born in 1946), or for that matter, he might have come home early, or never gone to fight in the first place (if the reason Hitler wins the war is that the U.S. never gets in it to begin with)–but in any event, the chain of events that leads to my father and then to me would have been broken. None of that means you can’t at least speculate about alternate histories; it just means that the habit of some fiction writers of having amusing little in-jokes like, hypothetically, having the American Quisling or the Leader of the Resistance (depending on the writer’s political preferences) in Nazi-occupied America being led by one “George Walker Bush” don’t make a lot of sense (since the real-world analogue of the Alternate History character in question was born in 1946 and is the son of a U.S. Navy officer, who, again, would have been a P.O.W., or dead, or otherwise occupied at the crucial moment that led to the younger George Bush’s birth. You could, of course, include a fictional analogue of George H.W. Bush in your “Hitler Wins the War” story.)