Rowling still has them all, and may decide to destroy them
In a recent podcast interview Rowling has done, she was asked about her manuscripts and she said that she still has them all herself, and that she's considered donating them to a library but has held back mainly because she isn't comfortable with sharing them.
Simon Armitage: Is there a J.K. Rowling Archive somewhere of notebooks?
J.K. Rowling: No, no, no. It's all- I've got it all. I was talking to Ian Rankin about this the other day because he's donated his papers to the National library of Scotland? I think that's right.
Simon Armitage: But you've got all yours?
J.K. Rowling: But I've still got mine, yeah. I'm not even really sitting on them, it's just I don't know, I feel- I don't know what I'm going to do with any of it. Part of me thinks I'll just have a big bonfire.
Simon Armitage: Don't do that!
J.K. Rowling: No, I don't know. You know Nabokov said that awful thing about people exhibiting their first drafts, "it's like passing around samples of sputum". That's one of the quotations I've got on my writing wall and I do feel like it's passing around examples of sputum.
J.K. Rowling w/ Simon Armitage on The Poet Laureate..., BBC Website, Extended Version (24 July 2021) [22:28 - 23:11]
There are very few publicly released pages of Rowling's manuscripts. These tend to come from Rowling's old (pre-Pottermore) website, from a few early television interviews she did, and from a few pages that she lent to exhibitions.
In particular, as it may be relevant to where (if anywhere) she eventually chooses to donate her manuscripts, Rowling has previously lent some pages of manuscripts to exhibitions at the following four libraries and museums in the UK.