I know it sounds like the set up for a stand-up routine, but it was a thought that occured to me while shampooing in the shower.
How do one handed people wash their hand - front and hack?
When I broke my wrist, I would wash the back of my non-broken hand by rubbing it against the side of the sink basin, or if in the shower, just rub it against another part of my body. The palm of the hand was easy enough to manage by enough contortion of the fingers, except for the heel of the hand. Again, I’d rub that against the side or rim of the sink. Drape wet and soapy paper towels over it if the cleanliness is questionable.
Crushed my left wrist in a motorcycle accident and had to wear a cast for almost a year.
I used a washcloth, set it in the bottom of the sink, soap it up and get with it.
Wasn’t all that difficult. 
My stepfather is missing his left arm from right below the elbow. He uses his “nub” (his term, not mine!) to do just about anything normal people do with two hands - including washing “hands.”
I’ve watched him do it often, and it really is just like the way two-handed people wash their hands.
I admit to wondering what people do who don’t even have a nub to work with, but he’s done a lot to change my perceptions of “disabled.”
Good point, I was thinking about “public bathroom” versus “showering at home”; didn’t even think about home sinks.
When I was wearing the cast on my left arm, I rubbed my right hand on my stomach to wash it.
Kinda weird in public bathrooms though, isn’t it?
or a Zen koan.
You can use water pressure from the faucet to do a good bit of the cleaning. Using hot water (well more than lukewarm, but less than scalding) obviates the need to lather up every square inch of the palm. Otherwise, it’s easy to soap up the fingers themselves, and the center of the palm.