I do wonder if in a timeline where there was no American Revolution, North American and British English would remain at least slightly closer together because there would be stronger cultural ties. British-American aristocrats might self-consciously echo aristocratic accents from the mother country, and the influence might start flowing in the other direction as well as the colonies rose to equal Britain in power. For the non-aristocratic, a larger percentage of the immigration to British North America might be from Britain itself, leading to more commonality among accents.
The thing is, they DID that in OTL. It still faded away/merged into other dialects such as the New Yorker or Pennsylvania dialect. Furthermore, nothing like that really survived in Canada, itself de jure British territory for a longer period of time. If you want to see that dialect survive, I think butterflying away the Second World War and/or American cinema as it stood in OTL could work. Not that I'd approve (then again, I'm a rabid reverse-snob