So, the UK really lacks good mid range - RPM based guns for Anti Aircraft defence. However, there is a gun ingame which was not only used by the Royal Navy but designed for use by both the US and Royal Navy.
The 3" 70 Dual Purpose AA gun. 

This weapon was originaly planned to be put on the Leander class Frigate in the 1950s but wasnt ready so an autoloading 4.5" mark 6 (similar to the Mk5 that's ingame but with faster rate of fire) was used.
The 3" 70 however was used on the Tiger Class Cruiser which entered service in 1959.

Now the 3" 70 ingame is the Mk23, however realisticly, the Mk6 was based on the Mk26 US gun, however the Mk23 was just the inital prototype trialed on USS Carpenter.

Now considering we have the Prototype gun as a gun ingame, surely it would work with the UK having recived them "as the war fought on" as per the game.

Now, some people may say "well whats the point in the US AA then" Well just you hold on!
In actual service the Mk26 had reliability issues while the gun was also large and heavier.

So my suggestion is the Mk23 be copied from the US for the British, be made heavier and require slightly more space and a slightly longer reload.

The US Gun specs:
Req space: 38
Weight: 125t
Reload: 1.08second.

So my proposoal for the UK gun:
Req space: 45
Weight: 140t
Reload: 1.20second

Justification:
In Royal Navy service the gun had issues firing at its full rate of fire due to issues in the turret housing. So they were often instructed to operate at below maximum rate of fire.
The turret the British guns were used in were heavier, although could track targets faster.

The Royal Navy currently lacks a good RoF AA gun, and although I personally love the 5.25, I think the 3" 70 could be better for lower levels and also self defence.

Thank you for reading! :D