Can you tell me if a young woman (19) has the possible early signs of cervical cancer, will she survive?

Yes, by all means. Cervix cancer would be very unusual in a 19 year old. It is not uncommon to see abnormal cells on apap test which can go away with time. You just need to follow up with your gynecologist so that any suspicious changes have been resolved with longer time or they may need to be treated by a gynecologist.