Wow how difficult for you. Sounds like she doesn't know what to do herself and is aiming her anxiety and frustration on you, and hoping you will fix what she doesn't understand.
Just a suggestion, tell her that you have heard what she's been saying and how worried you are, and if she won't make the appointment soon, you will, and you will even go with her to the first one to see how you can help, and do it!
She may well argue against it, but I bet if you are prepared to make the effort and go with her, and if necessary make the appointment, deep down she will really appreciate it.
I think her excuses are just that, excuses, because if she really is depressed, she probably has little idea as to why, and that is soooooo scary - it's like - how can I talk to him, he won't understand, cos I don't understand.
Even if you take her but don't go in to the meeting, (play it by ear on her needs), just knowing you are there will be a big support I'm sure.
Best wishes - I only hope you can find a solution soon.