IIRC part of the reason for slow firing was the cramped gun-houses, as I understand it the slightly updated and enlarged ones on the post-war HMS Vanguard saw a fair improvement. Of course that's not to say that there weren't other factors such as ammunition hoists and rammers/loaders but space was an issue apparently.
Tony Williams did suggest a fair while back that the British would have been better off concentrating on just the 4-inch gun in twin mountings and the 4.7-inch gun in the 4.5-inch's high-angle twin mountings with a re-designed shell rather than the plethora of calibres they had in the 1930s and '40s. I don't remember the exact figures but size and weight-wise I believe it would have been enough to fit two extra twin mounting onto the Dido-class, at which point it starts getting much nearer to the Atlanta-class.
A lot of work had gone into the Twin 4.7 mount in the early / mid 30s only for the purse strings to be cut before work had finished on making it a good HA Gun - Tonys Suggestion IIRC was that for a little bit more work / money it could have been made into a much better HA gun as well (as good as the 4.5 twins) and his ultimate suggestion was to use this 'superior' Twin 4.7s as a replacement for the OTL 4.5, 4.7 and 5.25 mounts where used - with the 4" Twin still used as OTL.
That being said two of the considerations for the 5.25" was that aircraft where flying higher and higher and in the 30s where it was beleived that "the bomber will always get through" it made sense to have a weapon system not made obsolete by aircraft that could fly above its effective engagement altitude.
The expected threat to the RN operating in a littoral environment was throught to be 2 and 3 Engined bombers.
While single engined dive bombers (Mainly JU87s) did prove to be a serious threat the majority of attacks launched at the RN in the Med were by twin and triple engined bombers.
Also with ship launched torpedo ranges increasing a better weapon for engaging destroyers was required as a secondary weapon for BBs (4.5 was too short ranged and 6" too large and slow firing and not very good as a secondary AAA).
Im not suggeting that the Atlantas would have been bad in any of these 'other role' roles but part of the success of the 5"/38 was reliable proximity fused shells (so 1943+) coupled with its higher ROF (helped by a very well designed shell room and gun house) as an AAA weapon this is not the preceived primary role of the class. It was to act as a scout leader for a Destroyer Flottila.
That it happened to be very good as a AAA Ship and avaialble as that threat emerged was a happy coincidence.
Handsome ships as well IMO