I teach full-time didactic and clinical in an associate degree nursing program. My strongest suggestion is to check with your state board of nursing. There are state policies that dictate educational requirements for nurse educators. For example, there are states that simply require that the educator have a higher degree than the program in which he/she instructs, so you could teach an ADN program with a BSN, a BSN program would require an MSN instructor, etc. On the other hand, there are many states that have a straight-forward MSN or higher requirement. The NLNAC prefers that registered nursing programs have MSN instructors or higher.
In your question, you said "Can a RN teach nursing students with a degree other than in nursing?" What do you mean exactly? You are graduating with a BSN, correct? Are you thinking of getting graduate education in a different field? If your state requires a Master's degree, it is extremely likely that it will require a Master's in Nursing. Nothing else will qualify you.
Also, you described your program as RN-BSN. Do you mean you are graduating from a BSN program and you do not yet have an associate degree in nursing or do you mean you are doing an RN to BSN program? I ask that because RN to BSN often is written as RN-BSN, and if you are just now graduating for the first time then you have no nursing experience, but if you already had your nursing license, you may have years of experience as an RN already. Certainly, you will need experience as an RN to teach, but it is not entirely necessary that you have decades of experience as a nurse to teach. Far more important is that you have an advanced degree and the ability to teach.
If you really enjoy teaching, you may consider doing some precepting in the clinical setting and see how that goes. What is your motivation for wanting to teach? Why do you want to teach BSN specifically? Nursing education is a poorly understood career field with a lot of false expectations. I may be able to provide some insight.
As an aside, I'd like to mention that you will want to work on your writing skills/grammar before pursuing a career in higher education. A nursing program will want their educators to be excellent communicators, orally and in writing, and it is likely that they would have you do a sample lecture before hiring you for didactic instruction. I realize this is an informal setting, however, people who display poor grammar and sentence structure informally tend to do so in formal settings as well. That would be a huge turn-off for a potential higher education employer.
Best of luck to you and congratulations on your upcoming graduation!