I've changed the main translation to a grown woman, hopefully people will like it better that way.
'She is an adult' would be just Hon är vuxen in Swedish, there's a bit of difference in meaning. I don't think there's anything wrong with an adult woman either so that's still an accepted answer. Possibly it's more common in the US than in the UK. For instance I found a Vogue article with the title ”I’m an Adult Woman, and I Call My Mother Three Times a Day” http://www.vogue.com/13469579/why-i-talk-to-my-mother-three-times-a-day/
Of course 'adult' is often used in medical texts too, but you all know that.
'a mature woman' is in most cases en mogen kvinna
Re "she is a grown woman" You probably wouldn't say that in the UK, it seems to be an American expression. "She is an adult woman", or she is a grown-up woman" and most likely of all, simply "she is an adult" or she is a grown-up" would be a more British way of saying it
To me (British, in case that’s relevant) “adult woman” sounds absolutely fine, with very similar connotations to “grown woman”. E.g. parents discussing a daughter who’s left for college: “She’s not a girl any more, she’s an adult woman.”
Edit: for those disagreeing, see e.g. this chart comparing usage in the Google Books British English corpus. It shows that an adult woman occurs roughly around half as much as a grown woman — less often, but quite often enough that it’s a genuine widespread usage not an error.