Glad to help if I can.
The underwater ballistics of spinning projectiles tends to be highly unpredictable. Very few tests were completed, and in those tests which were completed, the results were often problematical. Prior to the advent of modern technologies, about the only way to track a projectile in its underwater trajectory was via photography, and this ended up being entirely impractical due to the the obscuring effects of the impact splash along with the need to photograph the subsequent movement of a bullet that was often twenty or more feet below the surface, in two directions simultaneously, i.e. once from the top (for deflection) and once from the side, (for trajectory curvature). In practical terms, about the best that could be done was to fire the projectile 'short' so that it subsequently passed through a series of nets. This gave a rough idea of the depth of the shell at any point, and its deflection as well, although its orientation and speed often remained unknown.
A lot more testing was done during World War II, but unfortunately -- at least for those interested in gunnery -- most of these tests involved the entry characteristics of torpedoes and rockets, both of which are typically much longer than a spin stabilized projectile, and both of which typically enter the water at well under the velocities normally encountered with projectiles.
That being said, the general plan of the underwater trajectory is fairly well understood. After impact, most spin stabilized projectiles tumble rather violently, primarily because the spin rate necessary to stabilize them in air is several orders of magnitude less that that that would be required to stabilize in water. Often, after a short travel, the projectile restabilizes in a nearly 'base-first' attitude, rotated about 150 degrees from its original orientation. The high density of water means that projectile deceleration is also very rapid; as a 'first cut', the velocity at any point can be quickly approximated via standard momentum equations, i.e. by assuming the momentum of the projectile is transferred into displacing a given volume of water. In general, the trajectory underwater will take the form of a nearly circular curve, concave upwards, with the projectile re-exiting again at some point down range from the impact point if it retains enough initial velocity to do so. Curiously, the radius of the underwater trajectory is as well correlated with the relative density of the projectile and the water, than with any characteristics of its external geometry.
As might be expected, fuze action, as has been demonstrated in actual battle damage, appears to be somewhat unreliable. This is probably at least in part due to the relative unpredictability of the post-impact trajectory, and the associated inertial loads etc. expressed upon the fuze train.
Incidentally, the oft-seen and reproduced graphic of the Japanese 'diving shell' striking Tosa definitely does not represent a typical case, even for a shell designed specifically to dive. Although it seems to be a fairly common thought that the Japanese in some way 'invented' this flat-nosed concept, in reality it had been known -- and employed -- rather routinely at least since the turn of the century, with ships being equipped with special flat-nosed 'diving' shells specifically to attack submarines. Projectiles, even diving projectiles, only very rarely travel long distances in a horizontal mode under water. The trick is to balance the upward forces causing the trajectory to curve, with the decreasing velocity and to keep these two 'in balance' so that the projectile neither rises nor falls as it slowly 'runs out of steam'. This is very difficult to do, perhaps practically impossible when one considers that the designer actually has control of only three or four of the eight or ten variables that influence the physics of the situation. It appears most likely that either the Tosa diagram represented an extremely unusual trajectory, or that the Japanese were confused and connected the underwater hit on Tosa to the wrong projectile splash.
I have published approximate equations for the underwater trajectories of spin-stabilized projectiles and will not repeat them here. Keeping in mind that the trajectory to trajectory scatter is typically very large, i.e. that the equations are more precise than accurate, they nevertheless can predict the AVERAGE underwater performance of a projectile quite well.
Hope this helps...