I can think of three writers who could have completed the series fairly successfully: Diane Wynne Jones, Elizabeth Moon and Mercedes Lackey. My first choice would be Moon, who has an excellent command of plot structure and young adult psychology (see esp. her Vatta space-opera series). Any of the three would enhance greatly the role of Hermione, which would be fine with me.
Those are all very different writers from Rowling both in style and content. None of them would seem particularly appropriate for Harry Potter (and I like some of their works; I'd say the same about e.g. Tolkien or Bujold, to name two other authors I like; the style just doesn't fit).
Besides, both Moon and Lackey are Americans; I'd expect them to be ruled out on that basis alone.
I also question whether we'd have an official continuation; that sort of thing doesn't seem especially common, despite the Robert Jordan example. It would be extra dicey if there was a battle over the estate; I assume Rowling has all the best estate planning money can buy, but that doesn't necessarily stop legal wrangling when that much money is on the line. No one would authorize a sequel until the ownership issue was settled.
The market for Oz books was so good that after L. Frank Baum died, the publishers had another writer continue them almost without a pause. There are rumors that the said writer, Ruth Plumly Thompson, actually wrote most of the "last" Baum Oz novel,
Glinda of Oz (1920). Thompson wrote more Oz novels than Baum did.
However, a "franchised" Harry Potter novel -- "J. K. Rowling's Harry Potter by Horatia Hackett" -- would probably have the same problems that most continuations and fan fictions have; the new writer not having the same background as the original, and wittingly or not, introducing discordant elements, say, or the problem of playing to what the publisher or the core fans want, which may not be the best.
I tremble at the thought of
The Quest of Tauriel, a J. R. R. Tolkien's Middle Earth Adventure with all the trappings of contemporary fantasy.