To become a physiatrist, you must go to medical school first and then specialize in physical medicine. The other way to work in a rehabilitation setting, and do rehab with patients, is to become a Physical Therapist. That means, at the minimum going back for a Masters degree in Physical Therapy. To develop the requisite knowledge as a nurse would take too long otherwise. Your background in Exercise Science would give you a very good understanding as to why and how the body accomplishes what it does under exercise conditions. It does not get into injury physiology to the same depth that you would get in a Sports Medicine program or a Physical Therapy program.
As an exercise physiologist, at least at a bachelors level, you should have a better understanding of normal physiology than most nurses.
From everything that I have seen up to this point, nothing in nursing teaches you how to be a rehab nurse. There really isn't any background information/knowledge provided to the new grad nurse that would allow that nurse to function in a rehab setting, providing rehabilitative services, and being able to safely implement the rehab plan, let alone design a rehab plan, for each individual patient. An orientation to that setting would be fairly long because the new grad would have to learn all of that background knowledge, some of which you should have already obtained.
In my case, I would have that ability as I already have a Sports Medicine Bachelors, and 2 1/2 years of experience working as an assistant athletic trainer at the collegiate level, division three. So have a pretty good idea what is needed, although I have not worked in the clinical setting in the role of a rehab specialist, whatever specific title that may have such as physical therapist, occupational therapist, physiatrist, whatever.