The standard British destroyer weapon through the 1930's into the early 40's was the 4.7" open mounting. In only one class type did the Admiralty try to put this gone into a turret with the L & M. class. The biggest drawback with this weapon was the low elevation of the mountings. A high elevation 4.7" mounting was designed for the Nelson class but this was a large heavy mounting and with fixed ammunition it was not suitable for use on destroyers (or any other ship for that matter). The answer was to produce a dual purpose (DP) mounting of a size that could be used in all classes of ship from destroyers up to battleships and aircraft carriers. The Admiralty started work on DP weapons as early as 1931 and with a bit of thought could either have retained the 4.7" size or gone to the 4.5" gun size that we are familiar with today. The trials with the fixed ammunition on the Nelsons 4.7" should have put that option out of contention for any future weapons. Unfortunately the Admiralty tried the same fixed ammunition system for the 4.5" weapon developed in the late 1930's that became the BD/UD mountings on the Renown, Ark Royal etc with similar results. At that time the only way to go for light weapons of this size was the bag and shell separate arrangement.
I do not know how many zillions of 4.7" rounds the Admiralty must have had in stocks all over the world, but it must have been an awful lot. Keeping the 4.7" gun size for me is a no-brainer, all they needed to do was to produce a decent dual purpose mounting with 80-85 degree elevation that could be fitted to destroyers and other ships in single open mounts and twin light or armoured turrets for the capital ships. Starting development in 1930-31 the Admiralty could have had such weapons available in 1935-36 and these could have been fitted to the H (possibly) and I (definitely) class destroyers. Instead of the four single open mountings, two light weight twin mountings could have been mounted and the end result could have looked like the ship below.
The mounting I have used is big enough to take either a 4.5" or 4.7" high angle gun. Changes to an I class ship would be mainly in an increase of breadth to take the turret, a slightly longer hull to compensate for that and also to give room for the quad 2 pounder mounting between the funnels. From a 1500 ton ship, the alterations would probably have taken it to a 1600 ton ship and would have been a step between the A-G types and the following J design which I would expand to a 1750 ton design with 3 turrets.
With the drawing, removing the forward superfiring mount superstructure allows the turret to be mounted further aft to avoid having too much weight forward. The extra weight of the turret is somewhat compensated for by the removal of the mounting superstructure, which could have been removed entirely but as with the later HMS Savage I have retained enough of the superstructure to mount some light AA weapons.
Having had all the experience lately with drawings for the A-I series and the J, K, N, series of destroyers, finding an alternate was not difficult.
With the alternate J, I have kept with the 3 boiler arrangement to keep the Chief of Engineering happy, otherwise it is essentially a lengthened I to take the 3rd turret.
(Yes Kim I actually drew a real COA for Jupiter)
As an alternative to the turret mounted 4.5"-4.7" could be a more AA oriented 'H' class destroyer. But as noted elsewhere the Admiralty was worried about being outgunned by its foreign contemporaries. So the chances of a reduction in gun armament is unlikely especially as the Admiralty already had the 6x4" sloops in production.