What advantages would one get in reducing the maximum speed on a world war II line battleship to 15 knots? Longer range perhaps? Thicker armor? Less crewing and operating cost? Would overall construction cost be lower? What disadvantages would one get? It cant keep up with fleet carriers so it's no good as an escort. It can't dash to an area of operation quick enough. It will have to wage war at its own time and pace (!) Would its effectiveness as a fire support ship at land targets be impaired? Probably not. How about fighting in the line? Do battleships need to go at normal flank speed during a line battle? I read from somewhere that any speed higher than 20 knots makes accuracy past 10,000 meters very iffy. You could dash at 30+ knots only if you and your opponent are at a collision course (so your firing accuracy keeps improving as you approach each other at a certain combined speed.) In a rare one-on-one shootout with another BB, I reckon there's no serious disadvantage. High speed means you can position faster but that can be negated by other methods. You can outrun your opponent, but a battleship is supposed to stand and fight, not run away.
"He who fights and runs away lives to fight another day." The unit with the superior speed can usually dictate the engagement.
VS Nowaki the US battleships shot pretty well even at flank speeds. If you had a slower battleship you loose strategic mobility as well. Not sure it would have a huge impact on range. You should be able to get either a better armored or (inclusive) better armed ship for the same tonnage. Most of the battleship actions in WWII the speed of the various forces played a big part. In the Med it was important both for intercepting or avoinding interception and in getting away from air attacks. Certainly critical in Rheinbung and the battles around Guadalcanal not to mention Leyte.
Maximum speed of 15 knots?! What is that ship going to do? Catch kayakers unawares? Sink random pieces of flotsam as it drifts past? It's cruising speed is going to be even lower (say what, 10 knots)? The maximum speed of this theoretical ship is pretty much half that of any of the major WW2 battleships (Littorio-class 30 knots, Richelieu-class 30 knots, Bismarck, 30 knots, Iowa-class 33 knots, King George V-class 28.3 knots, Yamato 27 knots). Half. It's hard to see how the added size, guns / armour are going to make any real difference, when it's speed renders it completely ineffectual as a naval asset.
Look at the operational histories of the older WWI-era ships in WWII: the British R's, the old US BBs (Arkansas through Maryland classes), etc. These were designed with top speeds in the 21 knot range, only about 25% faster than your idea. Twenty-one knots was OK by 1914 standards but not by the standards of 1939. In particular, because of the speed difference the older battleships classes could not work effectively with aircraft carriers, the most important class of ship in WWII fleets. This meant that older battleships were progressively relegated to shore bombardment (in which they often did very useful work) and convoy escort. A 15 knot ship would have had even more limited value. You don't invest time, money, personnel, and slipway space in a large vessel which can only fill a small role. Such a battleship would be the maritime equivalent of the German Maus and E100 tank projects.
By reducing the weight and size of the engineering plant...All of th thee above, as the ship designers see fit. Likely, the war will leave it behind. It's to slow to work with the fleet. It's to expensive to for a bombardment ship (unless it is a monitor) or slow convoy escort. It's very vulnerable to air or submarine attack. It's too slow for a commerce raider. There is not really much that it can do. No. There are fire controls that take own ship speed into account. Mostly it is target speed and distance(longer shell flight). A faster ship will move further during shell flight, thus increasing the area a salvo needs to cover to obtain a possible hit - this is why most consider speed, to an extent, to be armor. A slower ship will remain in a salvo's CEP longer, thus is more vulnerable to being hit. IIRC, a 30-knot ship will travel 700some yards in 45 seconds, a 15-knot ship will only travel about 350 yards. Further, the slower ship will probably be slower to turn. A battleship is supposed to be able to sink it's opponent. If the British had 15-knot battleships, both Bismarck & Scharnhorst would have escaped. Had South Dakota & Washington been 15-knot battleships, they probably would have missed the Kirishima task force, if they would have been in the area at all.
I agree that 15 knots is too slow, but I also think the importance of speed is over-emphasized. Of course it comes into play if one side is seeking to avoid action, but battleships are usually trying to conduct - or oppose - some mission which would be compromised if they ran away. Situations like Bismarck being sent commerce raiding were the exception rather than the rule. You never get something for nothing, and speed is expensive, in capital ships around 2000 tons/knot. Consider examples like Queen Elizabeth and Hood or South Dakota and Iowa, ships of comparable armament and armor but the faster having to be over 200 feet longer and 10,000 tons larger. In the USN case, we might imagine a less dramatically stretched SoDak of 45,000 tons accommodating a fourth triple 16" turret, a significant increase in fighting power.
If you compare say the Colorado to the SoDak there are a couple of huge advantages SoDak has over Colorado. Fuel economy was one but the cruising speed was also very important. When you are conducting campaigns over thousands of miles a, say 20%, improvement in cruising speed can mean days in arrival time. Note that the USN carriers weren't accompanied by BBs on some missions because the BBs would slow them down.