First of all a huge congratulations on completing your nurse training. I can only begin to imagine the amount of determination and courage it has taken to get through your training, especially when dealing with depression and self harm.
In terms of what you do next I can't really answer that for you. From your post I get the impression that you feel you should take a small break before beginning your career. The 6 month recovery programme sounds like it could be a positive step, especially if you are struggling so much right now. Have you already applied if you're waiting to hear if you've been accepted? It could put you in the right frame of mind to begin your career after you've taken care of yourself for a while.
On the other hand, you see what occupational health say and start applying for jobs. You also say that when you were training you had purpose and focus. These same feelings could find themselves transferred into a job?
I know what it's like to sometimes feel hypocritical in a job. I'm a teacher and struggle with self harm and eating problems, but these never impede on my ability to do my job or to deal with pupils in vunerable situations. I know in your job you must experience this to a much greater degree, but you must have shown a good deal of professionalism in this respect in order to complete your training.
Sorry if I haven't been much help and it's a bit of a ramble!
...and yes you should be happy and proud of yourself