Those radars you have built up ever so well, are the fire control radar units for the 16" gun direction system.
While the individual turrets have "long base" stereoscopic rangefinders, those directors are for back-up, for engagements of only 10-11 miles or so.
The gundirectors up on the towers have shorter base stereo (this works by angling prisms at each end of the director arm until the target is aligned; trigonometry does the rest) but with significantly more height and a longer-18 to 22 miles--horizon line-of-sight..
Now, adding radar means being able to range targets in the dark or in poor/no visibility. Tracking 16" rounds with vacuum-tube technology aided by the dimensions. You can see 16" shells in flight with Mk 1 Mod 0 eyeballs. So, getting an RF signal off one is not that tough.
While on topic of main batteries, they are a unique operation at this scale.
This is a sophisticated dance. A target is identified. The gun directors give range, bearing, speed and direction of target. That is all entered into the fire direction computer (pure analog mechanical gears). This information is passed to the Gunnery Officer, who duly reports it to the Captain. The CO then decides on the tactical particulars. He then passes that information to the Gunno, who then communications that to the direction people. Who then plug in the data, and get back gun tube elevation and the train (rotation) angles for each of the engaged turrets. At the same time the gun crews are given the order as t which kind of shell to load. The turrets and gun tubes slew to the angles given by the directors. The tubes dip to the loading angle the rammers require, and bob back up to the shooting elevation.
Typically, two rounds are fired, to "range" the target (sometimes two per turret). The fal lof shot is adjusted until the target is "bracketed" with one long and one short. Said task being complicated by the target maneuvering, and likely shooting back. The results of all this being passed back to CIC. Once on target, Gunns, in the CIC will send to fire for effect, so the shot volume goes up quite a bit.
However, with the target maneuvering and your own ship maneuvering, you do not want to invest a lot of rounds on any given volley if the target is unlikely to be there during 60-80 seconds of flight time (at 20kt, that's 2000 feet traveled every minute).