Note: I realize that some of the secondary guns listed are dual purpose and some of them are not so I don't mean that it's a "well you're comparing apples and oranges" - it's more of a general discussion on logistics of ammunition types.
Edit: when I say secondary guns, I mean secondary guns for BBs and main guns for destroyers and cruisers.
US Navy:
Main guns of BBs from battleship not scrapped after WW1 to WW2:
- 14 inch - Standards
- 16 inch - North Carolina, South Dakota, and Iowa classes
Secondary guns on BBs or main armament on cruisers and destroyers:
- 4 inch
- 5 inch
- 6 inch
- 8 inch
Royal Navy:
Main guns of BBs is less of an issue as the QEs/Renowns/R-Class were WW1 vintage and the KGVs were matching the Washington Naval Treaty:
- 14 inch - King George V class
- 15 inch - Queen Elizabeth class, Renown class, R-class, Hood, Vanugard
- 16 inch - Nelson class
Secondary guns on BBs or main armament on cruisers and destroyers could be any of the below:
- 4 inch
- 4.5 inch
- 4.7 inch
- 5.25 inch
- 6 inch
- 8 inch
- Possibly forgetting others.
In terms of the Royal Navy, I'll just make the comment that when you are the biggest navy in the world (or one of the biggest at the end of WW2), you can afford to take on inefficiencies in logistics if it means you can kill your enemy better and faster. The American side seems to be more focused on standardization than the British - especially during WW2 where anything new that floated had a good chance of having a 5 inch/38 on it. But in fairness to the British at least for Vanguard, they used the same 5.25 inch DP gun mounts as the King George V class.