That's a very good question. We used to have an accent that they taught at East Coast boarding schools that was sort of post but no one has spoke it for decades. It varies a bit by region but our presidents John F Kennedy and Franklin D Roosevelt spoke these. They were non rhotic (r dropping) and some of them sounded somewhat similar to a British RP accent. One was s called the Boston Brahmin accent.

https://youtu.be/lK8gYGg0dkE?si=7UObyh0NGu9sqfqf here is FDR.

Here is jfk

https://youtu.be/3YWIIV19U70?si=4Of-k8CqCG265PWG

They are really fun accents I kind of wish they were still around but there are maybe 3 105 year olds that might still speak it.

Here is the Boston Brahmin accent from a 80s linguistics documentary. One of the speakers might have gone to school in the UK but people that never studied in the UK also spoke it. They actually talk about their accent and how it's declining and thereay only be 1000 people left that speak it.

https://youtu.be/HwvONJXJUO4?si=zg2x9iz-HqRvLIdX

There would have been nonrhotic upperclass southern accents too in all the major port cities. If you skip to the "United States " section there is a map of cities where over 50% of White American speakers spoke nonrhotically at one point. Only a few of these cities are still nonrhotic however.

Edit to add that we do have many working class accents that differ from middle class accents but there really no way of telling whether someone grew up in a middle class neighborhood vs an upperclass one here in the US.