If I understand your question correctly, you're asking why your page peel transition has a delay in starting and how to make it start faster.
Normally, transitions start on their first frame, if you check your transition in Viewer you can verify this, just go to the beginning of the transition, hold down the right arrow and you can see the progress. Shortening the transition will make execute faster but not start faster.
What will make your transition start slowly is a slow machine. The interval between the time touch the button and the time FCP starts playing is a function of your system's responce time. Anything you do to increase system speed will probably reduce the delay you are seeing.
In general, it is best to start the system a few seconds before a transition or effect to assess its playback.