I have taken the Gatwick Express (train) many times and they have the large toliets that you get on trains, but even in first they don't have power for laptops which I think stinks, but that is just me.
I don't think there is a bus service called Gatwick Express, though National Express does have services to and from there and they did have toliet facilities.
Don't forget to check ahead of time on the weekends, because they have engineering works between Redhill and Gatwick in the early mornings lately. You might find yourself sitting at Redhill having to use replacement bus service to Gatwick. National Rail has all the necessary information about closures and the like.