The term was originally used by opponents of the National Socialist German Workers’ Party and may have been influenced by Bavarian Nazi, a familiar form of the proper name Ignatius and used to refer to or characterize an awkward or clumsy person. The German form Inter-Nazi (shortened < Internationale INTERNATIONALE n.) which is attested much earlier may also have contributed to the adoption of the term Nazi.
But German Wikipedia says: ”Nazi war ursprünglich eine Koseform des Vornamens Ignaz, der in Bayern und Österreich häufig war.” I suspect the OED is just being stuffy.