Germany made the metric system the officially used legal system of measurements in 1871 after both the North German Bund and the South German countries had made their official system in 1868 resp. 1869. It also signed the 1875 Metre Convention that set up international coordination in the development of the metric system and the metric system had been in use in Germany ever since. So, in short, the Nazis used the metric system.
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The Third Reich used the preexisting metric system whose use in Germany far predated the Third Reich. The North German Confederation adapted the metric system in 1868 and it became mandatory for all German states with German unification in laws passed by the Reichstag in 1872.
The use of the metric system and the Anglo-Americans' own rather heterogeneous system of measurement created a degree of confusion during the war. Allied pilots that flew captured aircraft often had to adjust to metric instruments. In one case, Grumman's chief test pilot Corwin Meyer flew a Japanese A6M Zero on a test flight and he thought the aircraft was malfunctioning because the plane was taking forever to reach 10000 feet until he remembered the altimeter was metric.