First of all, the NASA noticed the destruction of the Russian satellite early on and even informs the crew about it, though just saying that it won't impede their operation. What they didn't anticipate was the resulting chain reaction with debris creating more debris and so on. This was indeed only noticed by the NASA when it was already too late (yet still only some 3 minutes after the satellite had been destroyed) and the debris was very near heading for the shuttle. It wasn't only the debris of the satellite directly that headed their way (and that ultimately destroyed nearly all the stuff in earth's orbit), but the additional debris created by the debris of the debris of the..., which is the reason why NASA at first didn't regard that one destroyed satellite far away as an immediate danger for the mission.
This kind of chain reaction catastrophe is indeed a real hypothesized scenario, called the Kessler syndrome. Yet I have no idea if it would happen exactly that way in reality, since the movie, while seeming pretty realistic to a layman, took some artistic freedom in many details. To get some insights into the particular scientific (in)acccuracies of the movie in certain parts, you can take a look at this interview (just don't take it too serious and don't let it ruin the movie for you, it's still just a fictional movie and not a documentary). It also gives some comment on the space debris:
What's your experience with space debris? If you're out on a space walk, what are you seeing around you?
Well, first of all, if you see it, it's too late. That was quite
implausible, to me, first off, the chain of events that would take
down the world's telecommunications. Things don't quite move in that
pattern. But that notwithstanding, if there were a cloud of
Micro-Meteoroid Orbital Debris, what we call MMOD, it would be coming
at tens of thousands of miles an hour. I don't know if you've ever
looked at a bullet coming straight at you. Hopefully not, but if you
had, you wouldn't appreciate it for very long. And these would be a
much tinier, swarmlike pattern of debris coming at perhaps twenty
times the speed of sound. And you wouldn't see it at all; it would
just rip through everything in its path. So it's kind of a
spacewalker's doomsday scenario.
So given that the movie wasn't entirely accurate in the way a hypothetical Kessler syndrome would pan out in reality, I think there isn't a sure way to say if NASA could have reacted a few minutes earlier or not. Yet in the context of the movie and given that nobody really took this worst case scenario into account, it isn't too unreasonable that NASA didn't take the destruction of a Russian satellite (which might not have happened for the first time) too serious before it was already too late (and also no way to say if the shuttle crew could have made it had they aborted at the moment the satellite was destroyed).