Sully 1996,
A nomination is the legal authority for the SA to offer an appointment. Without one, they have no authority to offer you a slot regardless of qualification or merit. That being said, you do not need an appointment to attend any of the service academy prep schools. In this case it would be NAPS in Newport Rhode Island.
Another thing to consider is why you want to go Navy. It is great that you have been nominated to the other SA's and even offered an appointment (congratulations on that, it is no small accomplishment!) however, as a current enlisted that has worked with other branches in joint operations, I can attest that life and careers vary greatly depending on the branch of service you choose. It is great to keep your options open and I commend you for deciding a life in the service, my last words of advice are to just be sure what you want to do and get as informed as possible. The Army can be a great choice for some and terrible for others. Same with other branches I imagine.
So, a NAPS slot is always possible without a nomination but very competitive. Whatever you decide I wish you luck and success as you begin an honorable service as a warrior and scholar. God bless