Hello!
I'm a student nurse, currently doing dual field Mental Health/Child Nursing, and I also have a mental health condition.
It really does depend on what issue it is and how it affects you on a day to day basis. On your university application they won't ask you about it, I did bring it up during an interview at Winchester Uni and they ended up offering me Adult Nursing instead of Mental Health. My recommendation would be to not mention it in an interview, I'm not sure if that had anything to do with them changing my course offer.
When you get accepted to a university and you accept the offer, they will ask you to fill out an occupational health form and you will have a few regular appointments with the occupational therapists, who will deem if you are fit to work. Plenty of nurses suffer from depression or anxiety due to work related stress, so there is no reason for them to say you can't be a nurse due to a mental health condition, unless it is severe and affects you from studying or completing placements.
I hope this helps (: