Not an expert but from personal experience I can share what we did during the Cape drought two years back.
Wash cycle water we kept for lawn exclusive as precaution, rinse and spin cycle water was used for all other outdoor plants. The grey water had no affect on any of the plants. We used normal retail washing powder, think Bio Classic at the time.
I guess it heavily depends on the soap/detergent/chemicals that you use in your washing cycle.
Some soaps are 100% biofriendly and are readily available but comes at a cost.