When is the best time to start getting in contact with the AMEDD recruiter?

The sooner the better. The AMEDD recruiter can give you a timeline of events. You won't be qualified until you have your BSN but at least you'll have an idea of the process and what's expected with you. Additionally, you'll have developed some rapport with said recruiter.

Now

Have you looked into ROTC? That might be a better option if you haven't started the program yet. Additionally, there might be scholarship money available.