Factual error: The bombing altitude for Nagasaki was 30 000 feet (~10 km), but in the opening scene the planes bombing Nagasaki are a lot lower than that. The bomb also detonated at 1,500 feet, not at ground level as shown. (00:03:25)
Factual error: In the Shinkansen fighting scene, the railway line is seen having color light signals along its way. But the Shinkansen uses a signalling system that doesn't involve the use of light signals at all.
Factual error: In the scene where Wolverine fights the ninjas on the bullet trains: 1) the shinkansen bullet trains are electric and have pantographs and overhead power lines - neither are visible. 2) None of the bullet trains have skylights.
Answer: The healing factor was only reduced. Like you said, had it been removed he would have bled to death and every time he used his claws he would be bleeding from his hands. It is not revealed if it is at full strength at the end of the film, but it is likely it will return to full strength over time.