Would Elba remain independent or would England or some other nation take it over?
I don't think it was really "independent". It was a British possession in all but name, really. Napoleon was only a nominal sovereign of Elba. It would probably be returned to France after a while, or maybe an Italian Kingdom or Republic (Tuscany?). Maybe the Brits would formally annex it and set up base there (it's very close to France).
Edit: Just from the Wikipedia article, it was given to Tuscany after the Congress of Vienna, which happened before Waterloo, so Napoleon was techincally still sovereign when it was given to Tuscany.