It seems as though they're still in her possession; other than a few items created specifically for auction, the only reference I can find to them being outside the possession of either Rowling or Bloomsbury, Inc. is when the manuscript of Harry Potter and the Philosopher's Stone was put on exhibit by the British Library in 2012.
However, the Harry Potter manuscripts are not part of the British Library's manuscript collection.
Rowling also posted pictures of some of her notes, including the original typescript of Philosopher's Stone, on her old website. Although that content was removed when Pottermore launched, the Harry Potter lexicon has archived it all.
Where might they end up?
If they end up anywhere, the most plausible location would be one of the United Kingdom's legal deposit libraries, of which there are 6:
But in practice, they could end up anywhere; J.R.R. Tolkien sold the original Lord of the Rings manuscripts to Marquette University, in Wisconsin of all places, in 1956. Of course, The Lord of the Rings wasn't the best-selling series of all time at that point, so the odds of Harry Potter winding up in one of the colonies is admittedly slim.
1 No, I'm not too proud to admit that I copy-and-pasted that