1940 Blue Jacket's Manual
Intrigued by the question of Naval small arms in 1940, I pulled out my father's 1940 Edition of the Blue Jacket's Manual, which was issued to all enlisted recruits and officer candidates. It was originally prepared in 1902, and revised several times by 1940. In particular, the 1940 edition was intended to "modernize the manual so that it agrees in substance with the latest training courses and includes such other information as would tend to make an able seaman and a thorough man-o'-war's man."
The contents on small arms would most probably reflect what small arms the sailor or officer would find aboard a typical naval vessel.
Chapter 40: "Carrying the Automatic Rifle and Submachine Gun" refers specifically to the Browning Automatic Rifle and the Thompson Submachine Gun.
Chapter 42: "Small Arms" refers specifically to the 1903 Springfield (not Model 1917 Enfield) described specifically as "the United States Magazine Rifle, Model of 1903," and "the Colt 45 caliber automatic pistol is the United States Navy standard." (No other small arms are specifically cited, except in short mention of life-line guns, etc.
Of course there is a full chapter on "Elementary Gunnery" and "Advanced Gunnery" addressing the basics and nuances of the large cannons -- 5" to 16".
There is very little about intermediate guns such as machine guns, submarine deck guns, or anti-aircraft guns.