Wolverine had healing powers from the very start. Before Wolverine had his skeleton enhanced with Adamantium, his bones were breakable as shown in X-Men Origins: Wolverine.
Did his healing power apply to his bones? If his natural bones were broken, would they have regenerated?
Yes.
Even humans bones regenerate at a slow rate. So Wolverine, as a mutant with healing factor-- his bones would regenerate, but at a considerably fast rate. As mentioned in a comment, without adamantium, his healing factor is even greater, as when it was in him, his body was constantly healing.
In the X-Men Origins: Wolverine movie, which you mentioned, he had his bone claws broken by Sabertooth before he had adamantium placed on his skeleton. It obviously grew back.
As mentioned in this answer, the The Muramasa Blade (the second one) is able to reduce Wolverine's healing efficiency.
Also see the question: How much of Wolverine has to be left to heal?
Yes, Wolverine's claws can grow back almost immediately, but they would grow back as bone claws and they don't grow back the same way. They would still be 9-10 inches long and the same width and length as before, but the bone structure or the pattern of the bone claws would be different. If you were to remove an adamantium bone then it would likely just regrow as normal bone, not adamantium, but there have been a lot of rumours that the adamantium does grow back, that it is beta-adamantium.
When his healing ability is not affected, he can regenerate his bones about 6 times slower than his muscles. Skin and muscles take 5-10 seconds, bones 30 sec-1 minute.
You need to chop off his whole head to kill him, like Spawn.
In X-Men Origins, everybody was asking the same question. The Wolverine (2013) showed the correct speed of his 100% healing factor.
his bones were breakable as shown in X-Men Origins: Wolverine- So did they or did they not regenerate at that point in the movie? Your answer should be right there...