Germany was faced with a situation whereby their heavy ships might well have to engage in close-range night actions in the North Sea. Given that there was no such thing as radar, and that Germany's escort forces were vastly inferior to a potential rival's, then the requirement for an effective anti-destroyer/light cruiser battery was strong. But such a battery would be too heavy to also be effective against aircraft. Hence, separate secondary batteries were required.
Germans had radar. It wasn't exceptionally good, because it was decimetric (Hitler's 1939 Directive cut funding for centimetric radar), but suitable enough for tracking (though ironically it still had a lower CEP compared to the guns of some ships it was installed on). As for anti-destroyer work, The US had shown that quick-firing 5-inch tier weaponry is more than a match for unarmoured/lightly armoured ships. A 6-inch gun is slower firing than a 5-incher, even if the latter's ammo is two-piece, while 3.9-inch tier guns are a bit too light for surface work. Another thing is that, with a single unified calibre, you only need one set of directors/calculators (since there's only one set of ballistic specs).
Most of the old ships had been built with casement before aircraft where really a threat and its really hard and expensive to refit turrets compared to just adding deck AA guns.
The DP guns are late 30s and cost a lot so only a few got finished in time for mass use in WWII (ie USN 5/38 who had time and money to spare) look at what happens to people who tried it too early on
http://www.navweaps.com/Weapons/WNFR_51-45_m1932.php!
Also remember that the DP guns that are actually good at AA are mostly lighter (max 5.25" and even then that's to big you want 55lb) than the good anti DD guns (any old 6" 100+lb) so its a trade off and as the relative threats change from night DDs to aircraft due to aircraft and radar improving the best gun changes.... and all heavy AA is very much weaker before US VT shells in 43+ anyway. That and the good or not so good fire control is one of the most expensive and hardest to make things to buy in 30s/40s.
The problem was that, on casemated ships, the secondary artillery was very limited in value. Poor angles and no real centralized direction meant they were mostly good for near misses, rather than actually sinking a modern ship.
Laffey was crippled by a hit from
Hiei's main, rather than secondary guns (after shredding the superstructure with its quick-firing 5-inchers).
Samuel B. Roberts and
Johnston were similarly mauled primarily by 14-inch shells.
The early DP guns (5.25'' KGV, 130 mm Dunkerque) had some performance issues not least because they still weighed towards the anti-surface side of the DP balance (too heavy and slow for AA work). The 130 mm's main problem was the heavy single-piece ammo (generally unworkable above 120 mm) although the quad mounts were very unreliable (owing to overly optimistic designers). Again, by the mid-30s the threat of nighttime DD torp runs had considerably decreased once radars were adopted (admittedly, the French, Italians and Japanese lagged badly in adoption). And, as I've noted, quick-firing lighter-calibre (4.7'' +/-) RPC guns were proven to be just as effective at opposing lightly armored ships as slower-firing higher calibre ones. As for VT fuzes, yes, that was the greatest single improvement to heavy AA, however linking an automatic fuze setter to a predictive target data calculator (who automatically updated based on radar returns) could provide relatively adequate results (well, as long as there was electricity) when combined with turret RPC.
IJN could not build any new ships from WNT (1923) to 2LNT (1 Jan 1937) without starting something very bad for them.....its rebuild that you can have or nothing.
RM could have but again politics would make it disruptive and I bet the RM sold the rebuilds like Super Hornet F18/E....don't worry its just a cheap modification not much more than a simple service...
The IJN got outplayed at Washington because the US had broken their cypher. Had they managed to keep a semblance of secrecy, I think they could have pushed for 'grandfathering' ships that were already around (or close to) 50% completed (IIRC the two Tosas and about half the Amagis), in exchange for accepting 'downgrades': the two Fusos and the oldest two Kongos would be decommissioned (maybe convert a couple to carriers instead of scrapping), the Tosas would have the same number of turrets as the Nagatos (sneakily keeping the 5th mount concealed under redesigned superstructure), the Amagis would use the 14'' guns from the decommissioned ships and have only 4 turrets (again, keeping the 5th mount concealed under light superstructure, and having the mounts capable of accepting 16''). Had they had the presence of mind to poison the conference by bringing up the difference between light and heavy cruisers (which happened anyway at 1st London), they might've been able to get away with their tricks in the resulting debate between US and UK representatives. On the other hand, the IJN itself was shattered into factions who displayed enormous mutual hatred, and it's hard to say whether the plan would've been digestible by some of the loonier admirals (though it would have landed them in the same 6-4 situation as post-WNT, except with newer ships).
The RM... I really don't get it. 12'' were already outmatched by the end of the First World War (even the French had switched to 13.4'' guns on the Bretagne). By the 1930s they were completely obsolescent. Slightly reboring them to 12.6'' was pointless; while on paper they were close to the calibre of the Dunkerques, the actual ballistics were inferior. And Italy could ill afford subpar ships, which the Cavours and Dorias definitely were, compared to the Littorios.