"Washing up" is mostly BrE, I think. I've heard it in films and such referring to washing the dishes.
In any case, "washing up" is rarely used in AmE. If you're going to wash clothes, you say "I'm going to do the laundry" or "My mom does the wash every Saturday." If you're going to clean up after dinner, you say "I'm going to do the dishes" or "I'll load the dishwasher, you go relax."