You are going to hate this answer ... but the truth is ... "It depends."
(particularly the better ones) do require a bachelor's degree in nursing before they admit you to a graduate program in nursing. They want you to have the theory and philosophy of nursing that is more prevalent in BSN programs than is often found in ADN programs. But there are probably some that don't require it. It will be interesting to see if others on this board can answer that question.
If you are thinking of graduate school anyway, and considering re-locating ... why not save time and money by going to a program that will give you a either a Master's Degree or a Nursing Doctorate all at once? There are some 3 year programs that will do this. It will be more expensive up front. But in the long run, you might come out ahead.
I've worked in leadership positons for a long time, and for career advancement, an ADN and a BS in another field is usually not considered as suitable as a BSN. For an entry level staff nurse position, the distinction is considered less important. But for career advancement, it is.