A fully analog/mechanical washing machine is just a big mechanical timer controlled relay behind the knob like a toaster oven. If you set it without power, it just runs through the timer without doing anything.
The machine would need to have a timer mechanism that is turned by an electric clock motor, not a mechanical spring with gears etc.
Or you could possibly set up an electric actuator that blocks the knob from turning. Then trigger both the power-on relay and the actuator retraction to do both at the same time.