Not the US south.
In terms of US-UK parallels, there's also the effect that geographical isolation has in making a sort of language "time capsule" where colonies stick with dated pronunciations or vocabulary. I've heard an attempted reconstruction of London accents from the early 1700s that sound a lot like General American or Canadian English. The same source had an early 1800s reconstruction that sounded rather Australian to my ears. One imagines that English people colonize either place at a given time, leave some trace of their time and place in speech patterns, and their relatives in England go on to evolve speech patterns in different ways.
Non rhotacism (lack of pronunciation of R) also left some traces in the American south (and northeast). That feature began in southern England after colonization, so I believe it was generally fashionable people keeping up with the latest English trends who brought it west.
That said, both AU and NZ have their own "TV accents", which in AU is referred to as "Cultivated Australian", which is relatively close to British RP (though not the same), and I expect NZ has their own equivalent, but most people would have something close to what you would generally think of people from that part of the world sounding like.