I'd say so, although I don't know how major a part of American culture it is.
It's pervasive. The 1939 movie is seen by the vast majority of children raised in the United States. Almost every American child has seen either that film or read or been read the original book for the last hundred years, really. It's a huge part of the fabric of American culture in that regard.
Certainly the film was a staple on the tv while I was growing up. And obviously this is a West End Musical not a play. We do West Side Story and the like so why not The Wizard of Oz?
Well, it's a bit different insofar as,
West Side Story may be a classic, but it's not something that
everyone sees in childhood.
The Wizard of Oz is right up there with traditional fairy tales, really.
And, yeah, I'm not saying that Britons can't do the story the same justice that an American cast can. I'm just saying, it's
such a defining aspect of American culture that I'm a little surprised it even appeals to non-Americans. I suppose there are plenty of Britons who feel the same way about
Doctor Who, though!