So, yeah, let's talk for a moment about the promotion system. In the TNG episode LOWER DECKS, you had three Ensigns all vying for a single Lieutenant-junior grade slot. The episode opened with Riker and Troi talking about how they just did crew evaluations only three months prior. In my world view, what was going on is what we call Below The Zone promotion whereby a sharp troop can be promoted ahead of schedule. (In the USAF, this is only available for E-3 to jump to E-4 six months early.)
So, in TNG, the commander of each large unit, and the Enterprise is big enough, is allow to bump one O-1 to O-2 ahead of schedule. Also, Doctor Crusher told Nurse Ogawa that she was recommend for promotion, too. I took this to mean aside form the other three, so perhaps Crusher is using the Starfleet Medical chain of command rather than the Enterprise chain / Captain Picard to push for Ogawa's promotion.
Beyond that, promotions should be automatic based on time-in-grade (O-1 to O-2 & O-2 to O-3), with commander's concurrence, of course, and then competitive against ALL personnel of the same rank & career path / specialty rating (for O-4 and above) across all of Starfleet, not just those on the same ship.