The three oldest battleships--Arkansas, New York, and Texas--were in the Atlantic until late in the war. Their prewar AA armament consisted of eight 3in/50s--not 5in/25s as in later ships--and eight .50 cals, added in the thirties. Despite their age they were among the first battleships to receive 1.1 quads, presumably because in 1940-1941 Atlantic Fleet ships rated the highest priority. By the time of Pearl Harbor all three ships had four 1.1 quads. New York and Texas, at least, had two additional 3-inch guns, for a total of ten. By late 1942, they each had eight quad 1.1s, and between 30 and 42-20mm guns; the .50 cals were gone, but the last of the 1.1s did not vanish until late 1944. ---Final armament for New York class: ten 3in/50; ten quad 40s; 44 to 48-20mm.
The other prewar battleships all received a battery eight 5in/25s during modernizations in the late twenties and thirties. Sometime in the thirties they also received .50 cal machine guns, usually eight. In 1940-1941 the King Board envisaged the replacement of both the 5in/25 AA and 5in/51 single-purpose batteries with 5in/38 dual-purpose guns, and the addition of quad 1.1s (later changed to forties and twenties). For the immediate future the battleships were to receive four 1.1 quads each. As few of these were available, most got four 3in/50s as an interim measure; of the Pacific Fleet battleships, only Maryland had 1.1s at the time of Pearl Harbor.
After Pearl Harbor the extent of repair and modernization varied considerably among these ships, and I won't try to cover all the permutations of armament in detail. AA modernization began immediately. Just about all of the serviceable ships got four 1.1 quads after Pearl Harbor, and 20mm guns were soon added as well. The New Mexico class were not at Pearl Harbor and not heavily modernized; all, for example, retained their 5in/25s until Idaho received 5in/38s in 1945. Light AA armament, however, was greatly increased. Idaho by February 1943 had a light AA armament of ten quad 40mm and 43-20mm. New Mexico and Mississippi got 40mm in late 1942 and in the course of 1943 the number of these guns rose from eight to twenty; they also had 43 to 50 20mm by early 1943, while retaining their 1.1s until late in the year.
Their armament remained at this level until 1945, when Idaho and Mississippi were refitted (the latter ending the war with 52-40mm). Nevada and Pennsylvania completed major refits in late 1942-early 1943. Both received sixteen 5in/38 in place of their 5in/51 and 5in/25 batteries. Nevada had eight quads 40s and 41-20mm. Pennsylvania had ten quad 40s and 51-20mm. This was apparently pretty close to the final configuration of both ships. Colorado and Maryland were not heavily modernized. Through 1942 each had four quad 1.1s and varying numbers of 20mm. By February-March 1943 Maryland had six 1.1 quads, 48-20mm, and ten .50 cal. By the end of 1944 each had about 36-40mm and 40-20mm. Maryland finally got 5in/38s in 1945 after kamikaze damage. Tennessee, California, and West Virginia were heavily reconstructed after Pearl Harbor. All got sixteen 5in/38s. They had ten to fourteen 40mm quads, and 43 to 80-20mm.
The North Carolina class were originally armed with four quad 1.1s and twelve .50 cal machine guns. This was increased in 1942: both ships received some 20mm weapons; Washington got two more quad 1.1s; North Carolina's 1.1s were replaced by ten quad 40mm when she was refitted after torpedo damage in late 1942. Surprisingly, the .50 cal armament was increased to 28 in both ships, but all were removed by the end of 1942. In course of 1943 the 40mm armament of both ships increased to its final level of 60. By mid-1943 each ship had roughly 46 to 64-20mm, although the numbers of this weapon fluctuated through the war.
South Dakota was the
only ship of her class completed with 1.1s (seven quad) and .50 cals (eight),
while her three sisters completed with six quad 40mm and 35-20mm. South
Dakota was in the yard several times in late 1942--grounding at Tongatabu,
the bomb at Santa Cruz, the 15 November surface action--and her armament
substantially increased; by early 1943 she had seventeen quad 40mm and
at least fifty-seven 20mm, a quite heavy armament for that stage of the
war. Her sister ships through most of 1943 had six to twelve 40mm quads,
and roughly 43 to 60 20mm. This was pretty close to their final armament,
except that Massachusetts was later upgraded to eighteen quad 40s.
---South Dakota, at the time of her famous antiaircraft action at
Santa Cruz, had four 40mm quads, five 1.1 quads, and somewhere between
35 and 57-20mm.
The Iowas as completed, or in immediate post-shakedown refits, were also close to their final configuration of light AA armament: twenty quad 40s in most, an average of about fifty 20mm.