Doctors are typically granted commissions right out of the gate.
In TNG, this apparently no longer holds true. Also, it may be that in TNG, doctors study medicine at Starfleet Academy. But TOS and its Abramsverse counterpart might well follow the practice of rewarding academic credentials with commission above Ensign rank. That McCoy would be a Lieutenant Commander the moment he boards the
Enterprise is still better interpreted IMHO as him holding that rank after having graduated as Lieutenant and then worked his way up the ladder whilst Kirk struggled with his graduate studies.
For how the authors might have thought of this issue, here's that alternate take on TMP, with the character Xon, who is stated to be a Lieutenant (j.g.) right out of the Academy: this leads Kirk to conclude that he must be "one of yours", one of McCoy's MDs, until it is pointed out that Vulcans also automatically graduate at Lt (j.g.) rank...
Carol wore a Starfleet uniform in STID, and was (IIRC) addressed as Lieutenant, so that tends to be a giveaway.
But since that was a sham, it might be that Carol's keel-hauling for stealing the uniform was postponed until the conclusion of the adventure, and in actual fact she was but a cunning civilian with access to daddy's uniform fabricator...
And I'm sure her father was an admiral in the prime timeline as well, as the divergence was only 33 years ago and he was probably already in Starfleet by then.
Since Marcus in the movie could well be less than 33 years old, t's not even certain she has the same father as her prime universe counterpart! The Abramsverse has plenty of central characters who are biologically distinct from their prime counterparts (and not just played by a different actor); theoretically, everybody younger than Kirk is different, and in practice at least Chekov certainly is.
Timo Saloniemi