How much do they know about your situation? At least here, managers typically get some training in sensitivity and confidentiality so we know how to effectively work with folks who are having personal issues.
I'd explain the situation to your GP during your appointment on Tuesday and see if they can't help you out with a simple letter. I honestly don't understand why they wouldn't if they know that you struggle with anxiety and depression.
Whether or not you have a letter, I would go to the meeting with the managers and just let them know about as much of your situation as you're comfortable with. Tell them you sometimes get depressed, but you have regular contact with your doctor(s) and genuinely want to work. Depending on what kind of job it is, they may or may not be able to have you back without a doctor's note, but I think being up front and honest is in your best interest.