Analog washing machines had one nice practical thing since you could force the "program" counter forward or backwards. This was especially practical if you were with a tight schedule and the program contained unneeded parts. You could skip them manually. Of course you had to know what you were doing, like not open the door with water in the machine.
It’s been a while, but didn’t the program knob in those machines turn in discrete steps? If so, then that system was — to be pedantic — a mechanical digital computer, not an analog one.
The most direct analogue (ahem) would be a music-box dial or perhaps a Jacquard loom.
The washer cycle(s) were driven by a clock which rotated a drum or cylinder with pegs that would start and stop specific actions. So, fill, agitate, drain, spin, rinse (fill, agitate, drain, spin), and spin-dry. The mechanisms were bog simple.
Whether you consider these analogue gear logic, or digital pin memory is somewhat arbitrary and a semantic distinction. Either way, the "programme" is fixed, and there is no interactive logic, only a pre-defined behaviour which is followed. Fill and drain were controlled via float switches, I believe.
Users could modify the routine somewhat by selecting different sections of the dial (which programmed different wash cycles) and by where within each the wash started (longer or shorter pre-soak), by selecting fill levels, and by selecting water temperature.
