First, I find I don't mind going a while between showers or other washing up especially in cooler weather. Yeah it sucks in hot weather, but in cool weather it really isn't that bad. Second, I am not really inclined to bike tour in real winter weather any way. Overnight freezing temps yes, but daily highs below freezing not so much. I am more likely to hike, snowshoe, or xc ski in winter if I want to do multiday stuff. That said those really fat tire bikes must be fun in the snow.
Also, I am fine with cleaning up with a wet washcloth when there is no shower available. It can be done with very little water if water is scarce, a plus when somewhere you need to carry water between distant restock points.