I don't have a Nautilus, but hopefully there would be a section in the manual dealing with how to clean the pieces appropriately.
I know with the seats I'm currently using, you can take the crotch strap out and run warm water down inside the female part of the buckle, and that is how that part is cleaned. You have to be careful not to soak the webbing of the strap part though. If your manual doesn't address it, call customer service and ask them to be sure.