Why gun turrets used different construction methods

Were battleship gun turrets cast, welded, or riveted?

All three methods were used; the choice depended on the manufacturer and the specific turret.

Heavy gun turrets did not share one universal construction method. Depending on who built a particular turret, its large structures could be cast, welded, or riveted. The method has to be checked against the manufacturer and design rather than assumed from the navy alone.