DoDMERB exam will be accepted by every SA, and ROTC boards. If the Army places in the order first, it doesn't matter, USNA will accept the results and vise a verse.
HOWEVER, you need to realize that you can end up with no DQ's for Army, but DQ's for the Navy (eyesight is the biggie). All DoDMERB does is qualifies the candidate, it is up to the individual branch to waive any DQs.
Same is true for SA and ROTC, candidates that are applying for an SA may find out that the medical issue will not be waived, but that same branch's ROTC will waive it.
Typically right about now is when the 1st of the applicants receive the letter saying go to these docs for the exam by X date.
OBTW one of the reasons they are waiting until they deem you competitive is the cost associated with the exam. You don't pay, but somebody does, and that somebody is the branch. SA/ROTC are not stupid, this is their own game of playing chicken. If USMA/AROTC orders the exam, USNA/NROTC will not be charged the cost, but they get the results. They know many applicants like you will apply to multiple SA/ROTC scholarship, why take it out of their budget if another branch will do it for them?