I have taken all kinds of cars with all types of engines to the carwash and washed the engine bay as a regular practice for over 30 years. The only ill effect I've ever had from this has been water in the distributor cap of a few gas engines such that I had to pull the cap and dry it out in order to get on my way.

I've washed engines this way when they were cold and when they were hot with little difference. It usually works better on a hot engine if you are trying to remove oil from a leak or a spill.

My $0.02,