To cancel the animation of evolving/upgrading Kai'Sa's abilities you have to upgrade the ability (clicking the arrow on the HUD) and, at the same time, pess B to recall (it can be done the other way around, but i feel like this way is better).

On Kai'Sa's Speedrun each animation cancel - except for the last one becuse the timer ends right when the yellow bar on your HUD disappears (just before the animation) - takes 2 seconds to be finished, which adds up to 4 seconds that you can save just by doing this.

Another thing you can do is to use TP and then evolve/upgrade your ability; the TP won't be stoped and you also won't do the animation. Same thing happens while in Kalista ult, Hexgates portals, being knoked up, etc...