Pure mechanical timer based washing machines have been driven by electric clock motors (line powered) for at least 50 years (as that is as far back as I remember knowing how they worked). So if line power is out, the timer does not run.
Of course, the delay timer on the line would need enough power handling capacity to carry the load of the washing machine.
My washer has dials to select the modes and then a final push button that activates an internal relay (guessing from the loud mechanical switching sound it makes when it goes on). If all you need to do is complete the circuit of the push button. You can just unmount that push button and hook its wires up to your timer mechanism which completes the circuit when the timer expires.
The dryer on the other hand does have a mechanical timer. But that doesn't activate until you turn the dryer on with the push button. So you could set the time of the dryer and not turn it on either, and use the same mechanism as above.
Mechanical Washing machines don't use spring wound clocks, they run of the AC frequency of the mains. It generally drives a small solenoid that increments the clock.
My current electronic washing machine saves its current cycle state when it looses power, and happily continues when power resumes 2 hours later (unlike my dishwasher, which does not). Although You'd have to start it before turning off the relay, so not as easy as the old mechanical dials
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.