I guess everybody can read what they prefer into the movie--it's nicely ambiguous on this point--but I don't think Spock and Uhura were a couple before the events in the movie. They were clearly more than just student and instructor, but I doubt anything could have prompted Spock to open up to her as anything more than an intellectual sparring partner.
He would certainly have been fond of her, but unwilling to really admit it to himself, never mind to her. His acute awareness of his split heritage and his desire to fit in as a Vulcan have driven him to perhaps seek an even greater depth of emotional detachment than the average Vulcan, if such a thing is possible. When he moves to Earth, it's more to spite Vulcan than it is because he wants to embrace the warmth of human culture. He still respects the Vulcan perspective more than the human one, perhaps even disdaining most human interactions as inefficient but necessary, to a minimal degree.
Uhura is intelligent, professional, and loves knowledge and learning, all characteristics that earn Spock's respect. She studies all forms of communication, although she's most known for her linguistic abilities. She's probably studied the subtleties of body language. Her knowledge of these subtleties and her extraordinary hearing (note her ability to detect Kirk's presence just by his change in breathing!) put her in the unique position of being able to "read" Spock's emotions / moods better than anyone, without him having to change his normal behavior. During the years that she gets to know him at the Academy, it is likely that a lot of what she would perceive when she was around him was his discomfort in his own skin. Given all of this, I expect that she would have felt rather protective of him. Out of respect for his wishes and to protect him from any potential censure, she would never have jeopardized their Academy relationship by hitting on him.
I can easily believe that he tutored her (where do you think she learned "all three dialects of Romulan" from?), and that would naturally have led to a stronger attachment to her than to any of his other students. He would have been aware of that attachment and it would have been distasteful to him (the only emotion he'd probably admit to regarding it). Even if he were to acknowledge to himself that he was attracted to her, he would never have allowed a more personal relationship to develop, because he wouldn't have wanted anything to cast doubt on the fact that she was advancing solely on her own merit. When he was required to complete the ship assignments, he chose not to assign Uhura to the Enterprise, to prove to himself (and to anyone else who cared) that he didn't have a special attachment to her. Uhura, of course, would have figured all of this out and that's why she starts out the boarding scene looking angry and makes a beeline straight for him. She demands that she be assigned to the Enterprise for two reasons: because she's proud enough to know that she's the best and thus should be on the flagship (the indisputably logical reason she gives him), and because she doesn't want to be separated from him. She's not ashamed of her attraction to him, is quite aware of it, and is willing to be very, very patient, knowing that it'll take a while to crack his shell, if she ever manages it at all.
I think only something on the level of losing his mother and Vulcan in the same moment could have:
- pushed him even as far as to storm out of the bridge, upset
- flagged Uhura on what was, for him, a emotional breakdown, and prompted her to break even her own rules about how she acted around him because she could perceive his pain so acutely
- prompted him to let her see him give in and then (!) to kiss her back, even briefly.
When Sarek (Spock's Vulcan father) later admitted to Spock that he loved Amanda (Spock's human mother), that was the real turning point, I think. After that, Spock would be willing to consider a lot more with respect to Uhura.
In other words, I wouldn't have swallowed Spock/Uhura without the extreme events in this movie, but now that it's perfectly logical, they've got me hook, line, & sinker!