He was no longer governor of the Bahamas by the start of the war. He and Wallace Simpson were living in Paris at the start of the war.
Here is the chronology.
Yes, the Duke of Windsor was living in Paris at the start of the war. He almost immediately volunteered for active service as a retired Army officer, probably in part because he thought it might help rehabilitate his image in England, but he may have also felt some sense of duty, and was appointed as a Major-General (as King he had been a Field Marshal, so this was technically a demotion) but as the highest rank he'd genuinely held (as opposed to one given solely because of his royal status) was Captain/Temporary Major during the First World War, it was a considerable promotion. He was appointed to serve on the British Staff in France, and it was at this time, some people suspected he may have leaked material to the Germans - personally I'm inclined to doubt it. It would have been treason and whatever his sympathies, that's a very serious accusation. Regardless, he was pretty useless as an officer (he didn't have the experience to match his rank) and it was for that reason, in May 1940, he was appointed Governor of the Bahamas. Basically to get him out of the way without overtly disgracing him.