Update:
Surgery actually took place this morning, and the surgeon said it went well. He repaired the damaged vertebrae, and he reported no perforations (that he could see) in the spinal cord.
They've put a HALO brace on, and my cousin's husband is supposed to be taken by air ambulance to a spinal clinic near Atlanta this week if all goes well.
I don't think any additional feeling or mobility has come back yet. It's still a matter of wait and see with the swelling. According to my uncle, the quicker he recovers sensation and movement, the better the long-term prognosis.
What he does have going for him is that he's young (early 20s), in good shape, and in good health otherwise. The doctors said that the damage would have been much worse if he'd been out of shape.
Thanks to all for the continued well wishes.