I think there's a lot of West Country in many (all?) of the English-language Caribbean accents, because that's where a lot of the plantation managers came from. Certainly, the Jamaican "axe/ask" thing is a feature of West Country English, along with "wops" for "wasp" and so on. Probably only among the very rural and/or very old these days, like a lot of these dialect features.I like the Bajan / Barbados accent because it occasionally sounds west country I think because of the way they pronounce their 'r's in the middle of words and the 'a's.
My accent falls in and out of a slight west country accent because despite having pretty rp sounding parents and extended family I went to bog standard state school so never really had any friends or acquaintances that spoke like that. I think it morphs and changes depending on the situation I'm in quite a bit too.