Why the Royal Navy split its dual-purpose gun effort

Could the Royal Navy have obtained a US-style five-inch dual-purpose quick-firing gun in the 1930s?

Yes, but parallel British developments diluted effort; the 4.5-inch between-decks mount matured before a suitable upper-deck turret.

Britain explored 4.5-, 4.7-, 5-, and 5.5-inch paths rather than concentrating on one American-style five-inch system. Henderson's 4.5-inch work produced a between-decks mounting for capital ships and carriers, while the upper-deck arrangement associated with later Daring-class destroyers took longer to mature.