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.
