Here’s a link to the program specifics. https://www.public.navy.mil/bupers-npc/officer/Detailing/aviation/Pages/NFOtoPilotProgram.aspxCan you be up to 29 when you apply to re-designate to NA?
As you’ve probably seen during your searches, I would not count on being selected for the NFO to Pilot transition. It is an extremely small program, and it requires you to have perfect timing (mostly out of your control), and great performance in your initial fleet tour. You’ll very likely only get one shot (if any chance at all) at the program due to your age (if you end up getting through flight school fast enough). As an example, I had fairly normal career progression (commissioned at 22), and was only able to apply once before I aged out.
I can’t speak to whether or not you can get a flight physical prior to OCS (I’m assuming no), but don’t get your heart set on transitioning after you enter the fleet. The odds are extremely low.
I’ll defer to @RecruitingGuru, but I don’t think you’re likely to get picked up for pilot if you go to OCS on an NFO contract, regardless of the outcome with your flight physical.
Bottom line: if you truly have your heart set on pilot, then you need to figure out a way to get the DP re-examined prior to OCS and get yourself a pilot contract, or explore other service branches.