While Americans do the laundry, the English do the washing. The washing up refers to doing the dishes. On occasion, you can still hear a dishwasher referred to as a washing up machine. The same terms apply to the detergents used during each chore. Laundry detergent becomes washing powder or liquid. Dish soap becomes washing up liquid or soap.
As for having a wash, I'm not entirely sure what this refers to as I've never watch anyone have a wash. It does not imply having a bath or a shower. It is frequently done at the end of the day before going out for the evening or before dinner or tea or supper or...that's a whole other post.