Looking at the circuit as I drew it, and I am only guessing I got it correct, I see it this way. When the main timer contact close, Timer 1 will begin its delay based on what it is set for, at this point nothing is happening. Following the preset delay timer 1 contacts that are normally open will close. Two things will happen, power will be applied to the transformer and timer 2 will be started for its programmed delay. Once timer 2 reaches its programmed delay then the timer 2 normally closed contacts that are powering timer 1 will open. The problem is things seem to stop there until the main timer contacts open and then close again to reset everything. I am just missing something in all this.
Also, I am not sure I read the picture correctly as to the circuit and what it should look like. The way this should work is when the main timer contacts close you want timer 1 and timer 2 to alternate on and off so you would water for whatever like 30 min and off for 15 min then on for 30 min as long as the main timer was energized providing power. You want timer 1 and 2 to toggle.
Later today if I have time I’ll mess with it. I have done timing like this at work but it has been awhile.
Ron