Here's an actual doctor commenting on the back break and recovery. His conclusion is...
- The diagnosis (herniated disc) is impossible given the trauma.
- You can't punch a herniated disc back into place.
- A herniated disc takes 8 weeks to 4 months to recover without surgery.
Another type of broken back is a compression fracture. They can be caused by a hard fall on your feet or osteoporosis. With no surgery, a compression fracture will heal in 8-10 weeks. Batman probably didn't have a compression fracture, but I throw it in there to show that a broken back doesn't have to be severe. Some people with compression fractures don't even know it, they think it's just chronic back pain.
Surgery takes longer to recover from, so Batman has that going for him. If I recall, it was easily several months between having his back broken and returning to Gotham. So the timing works out.
Being moved anywhere, much less across the world (probably roughly), is exactly what you don't want to do to a patient with a broken back. It greatly increases the likelihood of spinal injury. Points against for that.
As for the "treatment" by hanging, this is sort of like closed reduction. Sort of. If you squint. It's a fancy way of saying you put the bones back into place without surgery (the closed part), immobilize them, and let them grow back together. I guess the hanging makes sure his back is straight... maybe? Fashioning a back brace probably would have worked better. "It works best when it is done as soon as possible after the bone breaks"... oh well. The one part the movie definitely got right is a closed reduction is very painful.
We all know the real reason Bruce Wayne was able to recover from a broken back.