Early turreted secondary batteries

How quickly would turreted secondary batteries catch on if Britain fitted them to an early dreadnought?

Very quickly, because better arcs and more effective guns would also improve the path toward anti-aircraft defense.

A turreted secondary battery can point and train its guns more effectively than the arrangement considered in the question. That wider freedom supports better coverage around the ship. It also offers a stronger basis for anti-aircraft use as the threat develops. Other navies would probably adopt the layout quickly after a successful early example.