I have a soft spot for British accents with the same R sound as Irish and American. I guess that would be West Country, and also some versions of Scottish – not quite sure which ones or how widespread it is, but I did notice that the actress Karen Gillan pronounces R the "Irish" way, and she's originally from Inverness, so maybe it's western or northwestern Scottish. Or Pirate English. I wouldn't know.
Having said that, my fav UK accent is RP, even though it's technically not a regional dialect but a sociolect.
There appear to be a few different flavors, and I can't really tell them apart... there's supposedly something called "conservative" RP, and then there's "high" RP aka "The Queen's English" which I *think* is the old school Victorian era flavor of RP. And then there's a "stage" version of RP, some form of artificial RP (as if it wasn't already artificial) taught to and spoken by actors. And then there's different blends of RP and Estuary.
On the rare occasion, I turn against RP and become sort of like Otto in A Fish Called Wanda, like "oh, you British think you're soooo superior, don't you". Examples: I've noticed that politicians who speak RP have a propensity to pronounce words like issue and negotiations with a razor sharp S sound instead of "sh". So rather than "nah-go-she-ay-shuns" and "ish-ew" they say "nah-go-SSSSee-ay-shuns" and "iSSSS-ew". For some reason it disgusts me to the core of my being. Don't ask why, I wouldn't know. Another icky trait of certain flavors of RP (possibly "Queen's English" in particular) is when words that end with -y or -ie get this sort of twirl at the end that sounds like "eaugh". "Sorry" becomes "sorreaugh", "haberdasherie" = "haberdashereaugh", etc. It kind of sounds like they're throwing up in their mouths, as if they find it so loathsome to be surrounded by lowly peasants that they can't quite keep their breakfast down.
Finally – and admittedly, this isn't an RP thing per se but more of an isolated weirdness – my eyes roll every time I hear someone refer to Ralph Fiennes as "Rayf". Yeah yeah, I know, he's some kind of blue blood whose full name is Ralph Archimedes Bartholomew Poopington-Fiennes Upon Thwymesmansworcestershiresauce VIII, and they've been pronouncing it "Rayf" since the days of King Arthur or whatever... but honestly, I'm not going to pronounce an L as a Y. Why, because it's an L, insufferable fop. What's next? Pronouncing "Raymond Luxury-Yacht" as "Throatwobbler Mangrove"?