If you are treated with respect like you should, then no of course you will not get into any trouble.
All beds in the Psychiatric units have plastic sheets to protect their matresses. I very much doubt they'll say anything to you, they'll just change the bed and put clean ones on. Honestly you have nothing to worry about, you'll be fine.
All the other patients wouldn't even know Dale. The nurses go around making the beds everyday and change what beds that need new sheets etc, whether wet or not. This is usually done when patients are up and out of the way, in the day room etc. If some patients were to be around, to be honest, everyone's usually so caught up in their own problems anyway, I'm pretty much sure nobody will even notice.
I can understand you feeling embarrassed, but don't worry over what the nurses will think, they've seen a lot worse I can tell you.