She's his friend if nothing else; and he obviously needs her at those couple of moments.
That's what you do in a tense crisis situation. Abandon your post in the command center because your shipmate is sad.
Yes Trek crews have often been portrayed as unprofessional. If Uhura isn't capable of carrying out her duties when on the bridge she and Spock really should be on different ships. In my view, she should have been placed in charge of the bridge, then she would have had the authority to place someone else in charge of the bridge and leave her post. We wouldn't be whining about them placing Chekov in charge over Uhura then
It's not unusual for other crewmen to be given the more technical aspects of Uhura's job either. Spock often did so in TOS, which was annoying. At least she didn't spend the whole movie repeating the computer...