As I have said before (or if not I should have!), if Spock was so concerned about an obvious potential conflict of interest (a bit late btw) he should have excused himself from making Uhura's assignment.
I'm pretty sure their relationship was on the DL at that point, for the very reason that Spock is an instructor and Uhura is a cadet. Spock can't recuse himself from such a decision without revealing the relationship.
Spock said he didn't assign her to the Enterprise because "It was an attempt to avoid the appearance of favoritism"
If people didn't know about the relationship he wouldn't care about that, she would just be the best cadet getting the best assignment based on her skills.