I think it's Gordan who states the bomb detonates 20-some days after it's activated. That gives Bruce a little over 2 weeks to come back at 100% from a vertebrae popping out his back via a healing punch and magic rope.
The bomb does indeed take 5 months to degrade. The point at which Gordon, Tate, and Fox are talking about the bomb, there are only 23 days remaining and they need to come up with a plan. Batman has not yet returned from his exile at this point in the movie.
Exactly. Bane says explictly that the bomb will explode five months after it's disconnected from the reactor, in the scene down in the reactor chamber with Dr. Pavel.
Months pass, and winter comes along, while Gotham is occupied . . . and Bruce is stuck down in pit.
(Trust me, I actually had the entire story plotted out on a calendar so that I could keep track of how much time had passed between each scene! And I hid the calendar at the end of every working day!)