Regia Marina used triple turrets already in WWI and from a turret mechanism viewpoint they were quite well proven and gave little problems during WWII. Littorio class battleships triple turrets were quite reliable, obtaining very good outputs in action (e.g.see Gaudo action where Vittorio Veneto fired 29 salvos with an output around 90%), but they had a low RoF that "helped" in some way the "reliability" of the mechanisms and of the crew....
The problem of the Italian battleships armament (the heavy cruisers are another story) was excessive salvo dispersion, due mainly to exaggerated tolerances in ammunition specification/provision from national industry. The tests with (good quality) ammunition showed a very limited dispersion. The problem was only partially solved during the war, increasing controls on ammunition providers.
French turrets, as far as I know, were reliable too, but guns suffered even worse dispersion problems due to the twin of them being too close one to the other when firing together (as per Italian heavy cruisers). Also the outbreak of war with the ships not yet completed forced to use non-high quality ammunition (on Richelieu class I heard that Dunkerque powder was re-bagged to be used on them).
Main problem was (again partially) solved by introducing a small delay between one gun firing and another.
I think the same solution was implemented by design, since the beginning, in the triple German turrets (e.g. Scharnhorst class). Whitley (I think) speaks of a 10-20 milliseconds delay between the center gun (fired first) and the side ones..... I don't know whether someone (more expert than me....) can confirm this point.
British turrets problems were more linked to turret loading mechanisms themselves, due to British doctrine (correct, IMO) to increase RoF as much as possible, that imposed to the triple and quadruple turret (complex by design) mechanisms to be very "quick acting" while they needed to be dimensionally limited and required long periods of time to fix them.
I totally ignore American and Japanese gun turrets performances in terms of dispersion, RoF and reliability.....