That doesn't seem that striking a similarity to me, Beth. Unlike Winston Churchill, who insisted on 'narzy', Tolkien could pronounce 'nazi' properly, which makes it sound completely different from
nazgūl. A closer real-world connection is that between the Gaelic word
nasc, 'ring', and Black Speech
nazg, 'ring', on which connection
I posted some time ago.
Of course, in terms of the narrative setting,
nazgūl is simply Black Speech for 'Ring-wraiths'. I'm very sceptical of any reference to the National Socialist German Workers' Party, even though Tolkien did express a profound dislike of several senior Nazi figures. When I get back home tonight, I'll try to find out when the term
nazgūl emerged, and see if it corresponds to political events in contemporary Germany. I have a feeling that it won't.