For this Australian, I've encountered the word mensch in US media (TV and movies mostly) a number of times; each time I think the speaker was Jewish. The general meaning was always clear enough from context (to mean something along the line of 'a real human being', 'a good man', generally relating to a person who is consistently displaying qualities like generosity or responsibility).
I've also been called a mensch by an Australian colleague who is Jewish; while the word is not really used in broader Australian culture, in my case at least it was understood in a general sense.
Is "act like a mensch" too localized for ELU readers?
I think there has to be some level of tolerance for terms that are somewhat localized; I don't feel it's too localized, but is there any objective basis on which to assess such a question? I doubt it.
Please label your answer British or US.
Those are not the only varieties of English! I can't label my answer with either term, since it doesn't apply. While I comprehend idiomatic English of both varieties quite well, I don't speak British English, nor do I speak American English (though most Americans I encounter when I am in the US can't seem to tell that I'm not English unless I talk like Steve Irwin; when I am in the UK people seem to instantly know exactly where I am from).
There are more speakers of English as a first language in the Commonwealth outside Britain than in it, so I don't see any particular reason to accord primacy to British usage within it. On the other hand there's a degree of similarity in many varieties of Commonwealth English, and to the extent that there is, it may make sense to refer to that; with that in mind, outside the US, mensch would be relatively rare, and where it is heard, mostly via US cultural influences.