Wagner calls Woden, Wotan, by the way. I think he changed all the names - they obviously were not German enough. I did once read something about the epic poem Niebelungenlied expecting it to be pure Wagner but the story is rather different so obviously Wagner just took some names and ideas and made up his own story. But then, isn't the Niebelungenlied itself a condensation of other parts of Norse mythology?

So neither Tolkien nor Wagner produced stories particularly close to the original myths but they both relied on many of the underlying ideas in those myths. You can see where at least some of the inspiration came from.