I have absolutely no idea about the authenticity of the watch but the speculation is leaping far ahead of the available evidence. Airfield commander, someone working at a high level—we’ve seen this kind of entirely unfounded excitement build before. Not trying to be a killjoy, but a healthy majority of items with supposed wartime markings turn out to be counterfeit.
What if it’s authentic? Then an actual Nazi who loved Naziism so much he had a swastika engraved on a nice watch owned it. Hard to enjoy it, even as a historical artifact. What if it’s not authentic? Then someone desecrated a nice watch with Nazi shit. Why would they do that? No innocent rationale comes to mind. Why would anyone want it, then? Hard to answer that question, I think.