I remember now, that 20 years ago, when I was regularly hiking the Alps, I always had a bottle of water with me and I would just wash myself (and then my hand). I’ll think about returning to that method again.
I guess in a normal toilet setting water can still be used to clean oneself and then the cloth could just be used to dry. Should not get too dirty/stinky then and can be comfortably collected until being washed.
I like the idea!