You're assuming it's infection based. The original Zombie concept was magic/voodoo. And even if it is infection based he may become infected for a time until his mutant immune system catches up.
Then, if you think about it, the next time the Zombie Wolverine withdraws his claws (Ermahgerd, that rimed!), won't he just loose them? Lol, I mean there will not be any skin to keep them in place
Maybe. Depends on how far along the decay is but I don't think it's the skin that's a factor. All bone is held together by ligaments and moved around by muscle and tendons so at some point when his flesh totally decays, sure, he'll just fall apart. Haha.