This one is closely related to The Gutter Chain dishwasher.
I live in a rainy environment where you can pretty much count on some rain for 6 months of the year.
Most roof water is brought to the ground in downspouts. I prefer the chains that are used on many Japanese buildings. They don't clog up with leaves.
Provided that we have a suitably strong chain and lots of rain , clothing can be washed simply by hanging it on hooks that are placed along the chain. Large items like pants and shirts are given their own hook. Smaller things like socks, could be placed in baskets at the bottom of the flow. Sides of the basket need to be high enough that the socks don't go splashing out onto the ground.
The water is not heated and no soap is used. Still I know from my other experiments that clothes can become quite clean when hundreds of gallons of clean rain water are allowed to flow over them. Most of the dirt is gone after half an hour in a good rain.
I have done this in my stream, but during heavy rain, the flow can be tremendous. Some pants had to be retrieved from the valley below. I've also done this with a small cedar tree. I cut the top off and cut off all the branches, leaving 6 inch stubs. Dirty work clothes were hung on those stubs and left out in the rain. I returned to the farm during a dry day, to find my clothes nice and clean.