Water level switch is OK,timer switch is OK, but there is no sound at all from the agitator when the wash cycle should begin.If the timer is advanced to the water discharges and the spin dry cycle it works.

If it pumps out okay and spins okay, listen to see if you can hear the motor run in the wash cycle. If so, you probably have a stripped out block on the transmission where the agitator fits down over it. Replace that block and you've fixed it.