When and how does bat speed happen?

This study measured the kinetics of bat speed in 99 amateur players with an average age of 19 years old. 

Bat angular velocity and bat head speed increase simultaneously around the half-way point of the swing (this looks like the time of when stride foot or heel lands, but is not specified in the study other than the image below) and they increase together until around the 80% mark.

The angular velocity is then converted to translational bat head speed into contact.

An interesting finding was that torque created in the X-Y directions did NOT have a significant relationship to bat speed.  The authors cited that torque exerted by the wrists is small compared to the amount of bat speed that gets created. 

Summary:

The document is a scientific study that aims to identify mechanisms to increase bat head speed in baseball batting. The study recorded the batting motion of ninety-nine amateur baseball players and analyzed the kinematics and kinetics of the bat. The results showed that the bat's rotational power increased with the torque exerted on the bat's grip, but declined just prior to impact. On the other hand, the bat's translational power increased just prior to impact. The study concluded that the bat's energy, by the application of rotational and translational power at different times, contributes to an increase in bat head speed in baseball batting. The study highlights the importance of increasing bat head speed in improving the performance of baseball batters.

Link to article 

Horiuchi, G., & Sakurai, S. (2016). Kinetic analyses on increase of bat head speed in baseball batting. International Journal of Sport and Health Science, 14, 94-101.