...Maybe as a pocketwatch, it did. But since the case (along with any military markings that were on it) has been destroyed, there is no way to prove it.
Omega made a lot of watches under contract for the Allied forces in WWII (UK, US and Australia). The only Omega watches I've seen for the other side were personal watches that were engraved with commemorations of service.
Hope this helps,
gatorcpa