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.