Can I alter the 'timing' of my toilet flush?
Actually, that's how a toilet is supposed to work.

You can simulate a flush simply by dropping a couple gallons of water in right away.

 

The spray paint fill is the natural water level.  Flushing occurs by suddenly dumping excess water in the bowl.  This raises the water level at the back, an a siphon effect comes into play with pulls the remaining water out of the bowl, giving the typical whirlpool effect.

If you're not getting the whirlpool, and a clearing flush, there are two possible scenarios.


The tank isn't full, and not enough water is coming down to clear the bowl
There is a small blockage in the drain.


But timing isn't an issue.