When you apply to UC, and declare you are unable to work, you have to put a diagnosis in the box. You will also have to do the same when you fill in the Work Capability Forms.
The initial application for UC is online, but until you have a WCA, you will have to supply sick notes to the Job Centre and attend regular appointments. If your GP wont give you a sick note, you will have to look for work.
This^
I have just helped a relative claim UC and PIP, he has severe mental health problems (psychosis, clinical depression, ptsd etc.) He is a danger to himself and others when he gets very ill so it's not safe for him to be in a work environment. It was relatively straightforward for him to receive LCWRA but he has lots of medical evidence. Psychiatric reports from community mental health team, CPN letters, gp letters, letters to say he is prescribed antipsychotic medication etc.
My point is, without a lot of medical evidence you will likely be found fit for work. You need to write down all your symtoms/ difficulties (it's easy to forgot stuff when you're talking to gp) and book a gp appointment as soon as possible. You really need to be started on medication if you're at the point where mixing with people makes you ill. As pp said, as soon as you tell UC you're unfit for work, they will ask you for fit notes/ sick notes so it really is important you get to the gp straight away.
Good luck!