Call the school/exam place and explain the problem, see how they can help; I assume you have your "paperwork" telling you when the exam is, etc. Do you live at home with a parent? See if they can come with you and certify that you are who you say you are, first proving they're your parent, etc.

Any of your previous school teachers you know well enough to contact who might be able to call the school/help you? Did you have to have references for your application to nursing school? One of them can certify that you are you?

But call the school and find out if they can be helpful and take something else in your circumstances, even if you have to bring a boatload of other documents (birth cert, personal pictures with you and your parent together and your parents' driver's license, etc.)

Call DMV and make sure you can't go down there today and get a new one on the spot (to heck with mailing!). Do you have a passport? That would do.