Different schools work differently. In some BSN programs, a student enrolls in the nursing program in the first year and takes four years of coursework, graduating with a BSN degree. Some BSN programs are upper-division programs only…meaning that you take prerequisites (sciences, etc.) in the first two years and then formally enter the nursing courses in the third year.
It sounds like the program at your school is the latter type–a two-year upper-division program, and it’s taking you longer before you can transfer in because you decided on nursing in your sophomore year? But if so, why would it take another three years once you start on nursing coursework?
At this point, if you transfer to an ADN program, it might theoretically take you two years instead of three…but that’s assuming that the programs where you are aren’t waitlisted, that they accept all of your coursework, and that your completed coursework includes all of their prerequisites. Your best bet would be to examine the pre-nursing courses requirements from schools you are interested in and see what you have versus what is required. Then see what is transferable and what is not and go from there.