Yes, all body parts must have an owner in English, both AE and BE; they don't just wander around freezing unattended.
There are many synonyms for Hintern, any of which could probably be used in this phrase, though our shivering northerners are the ones who should confirm that. To me 'butt' is the most common in AE (BE 'bum'), and is indelicate but not exactly vulgar, or 'ass' (BE 'arse'), which is more vulgar but also very widespread. On a milder level, suitable for Grandma, also 'rear end,' 'tail,' 'buns,' 'tuckus,' whatever.
My impression from the previous discussion is that our British friends may tend to freeze other parts of their body off, namely balls or bollocks, only one of which is in LEO so far. Not sure what the most common non-vulgar alternative there would be, or whether British women ever get that cold. (-: