That's another thing about the movie...the discipline of the officers is atrocious even for Star Trek. Not just Uhura abandoning her post like you mentioned but everyone standing around watching Spock and Kirk fight and not calling security or intervening. Then the obvious Kirk moment of him assuming command...Bones' over dramatic statement of we have no first officer and no god damn captain or however order it went. Um last time I checked Bones you are a Lt. Commander...unless those stripes on your sleeves mean nothing, Sulu is a Lt, Uhura was promoted on the spot and is a Lt she could take command. Instead everyone is waiting around...until Jim sits down and declares himself Captain. Don't get me wrong...I love this movie...there are aspects about it that bother me though.
Also why are the Romulans using alternate timeline pistols and blasters when they should be using their 24th century disrupters? They should have used Prime Trek sound effects, unless of course those weapons were confiscatd by the Klingons when they were boarded and taken to Rura Penthe.
Bones is a medical corps (staff) officer, not a line officer.
As for the Uhura question, protocol gives way to plot conveniences.