Your fingers wrinkle because there is a thicker layer of epidermis there. Dead /dying skin cells absorb the water and expand and because the underside of this layer is attached to the more sealed, less-absorbing part of your skin, it wrinkles on the surface. Your toes and heels wrinkle a bit too for the same reason.
I never really noticed the cold vs hot water difference, so I’d have to guess that warm water absorbs quicker, kind of cooks into the dead skin, puffing it up and softening it. Warm things are more commonly reactive than cold, so that’s my guess.