I have been a CNA for awhile now (a little over 5 years) and I know that in my area when hospital jobs open for CNA's there is a stamped! The work is better, the pay is better, the hours are usually better, the benefits are better. So although the hospital is willing to train people you may be competing with experienced CNA's. Moreover, most CNA's (in my area-western US) have to work at least 6 mo in a nursing home before a hospital will even look at their app. So if there is a hospital that is willing to train you then apply!
You can put on your app that you are going to be starting nursing school
in the fall. This may give you an edge over other applicants (even experienced CNA's) as the hospital will probably want to hire and retain you so that when you are a nurse you will work for them. Many hospitals offer tuition assistance ect, in return you usually have to work for them for an agreed upon amt of time after graduation.
I will be starting an ADN program in the fall and I think being a CNA provides a lot of valuable exp. You get use to stinky smells (lol), seeing people naked, in pain, getting yelled at, not knowing what to do at times. You also learn how to take vital signs, perfom basic care to include bed baths, making beds with pts in them, charting, counting I & O's. You learn some medical terminology
(if you haven't taken any in school) or an expansion on what you already know, you get used to interacting with families, other medical staff, ect. Some hospitals will (the one I worked at did this) train you to do blood draws, EKG's, check blood sugars, ect. Overall working in the hospital as a CNA offers exposure to a variety of experiences and I think it will help me when I start nsg school this fall. I think you should apply to the hospital! I will keep my fingers crossed for you!