As others have said the staff in these places have probably seen it all before. However in my experience, some of them can be, shall we say, somewhat less than tolerant of patients with incontinence issues, while others seem to take a fiendish delight in ridiculing them, sometimes with little or no regard to the patient's feelings or their right to confidentiality.
I can say this because I have had first-hand experience of it. I was admitted to an NHS psychiatric hospital a couple of years ago, because I went through a very dark time in my life and had tried among other things to self-harm (cutting myself), and also seriously overdosed on prescription meds. When I was taken to the unit, I was to all intents and purposes unconscious; I was utterly unaware of what was happening to me or where I was. Despite the fact that my incontinence issues (mainly bedwetting) were documented, and also the fact that I had (without knowing it) wet myself in the ambulance on the way to the mental hospital, the staff still put me to bed without putting me in a nappy with the inevitable result that I wet the bed.
Later the following day, one of the "carers" (the inverted commas are my way of indicating that the person did not deserved the title of carer, since she seemed to care little about anyone) loudly announced to anyone who cared to listen that I would get on well with one of the other patients, a woman named Sarah, because both of us were a couple of big babies incapable of keeping our knickers dry, and walked away laughing. Myself and a couple of others put in an official complaint about the incident, though whether she was reprimanded or punished in any way I have no idea. Suffice it to say that nothing like it ever happened again while I was in there.
I never did work out whether the other girl, Sarah, was either actually totally medically incontinent, or simply too lazy to bother going to the toilet or even attempt to control her bladder. Whatever the reason, she was almost permanently wet, but refused categorically to wear a nappy. She simply seemed to just let go whenever the urge hit, uncaring as to where she was or what she was doing.
As far as I was concerned a member of staff checked every night to ensure that I was wearing a nappy before I went to sleep. I was also woken promptly if anyone noticed that I had fallen asleep in a chair, in the patients' lounge for instance. Of course this had the inevitable result that everyone, staff, patients (and even some visitors) knew that I was a bedwetter, even though that fact bore no relation whatsoever as to the reason why I was in there.
So if you do at any time in the future find yourself admitted to a mental facility, be assured that in all likelihood you won't be the only person in there with incontinence issues (real or otherwise) and that most if not all of the staff have dealt with it before. And hopefully with more sensitivity and compassion than was the case in my experience.