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Yes, it still does mean "buttocks" in American English and "woman's private parts" in British English, where the term is apparently considered quite vulgar. Fanny for buttocks is old-fashioned, like something your grandmother would say, telling you to sit your fanny down and stop making noise.
It definitely does not have any aristocratic vibes for me or for most people.
I would say avoid it even as a nn.

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Steve Martin