I would say that, even though Tolkien originally wrote it as an English mythology, it has so many elements that are applicable to everyone, really. Tolkien was, of course, adamant about the applicability of his book, so it was (one could argue) appropriate for anyone interested in a good story.
I would still class it an English icon, really. I find it hard to imagine any of the characters with anything but an English (or at least British) accent of some kind. Especially Gandalf. As Bęthberry has said, the Shire is very much alike to the rural England in which Tolkien grew up in and loved, and gives it a greater English feel.
What is more, it follows the stereotypically English home comforts; a comfortable home, a bright garden, toast in the morning, and, of course, a good pipe. The idea of the smallest people struggling on in the face of dire odds is an international ideology (I think), yet triumph over adversity is always a well-loved tale in merry old England.
I'm not saying The Lord of the Rings is not meant for any other country, but I think it can fit as an English icon. Even more so than King Arthur (him being French, of course,
) ... (Not that there’s anything wrong with being French, of course
)