At this point, there is no canon answer for this question. As far as I know, none of the Michaelsons have ever been seriously injured in the manner you're describing. As these shows are not really known for their strict and consistent canon, until it happens on-screen it would be difficult to even speculate.
However, if we take the show at face value then we have to assume yes, they would eventually heal from this. The primary reason is, none of the people who are actively trying to kill them have tried to do this to them yet.
In particular, we know of at least one person who has both the skill and the desire to try this with one of the Michaelsons -- their father Michael. He spent his entire life hunting them down, but more importantly, hunting down the white oak stake that would allow him to kill his children. He wants to get rid of all vampires, and killing one of the original line would wipe out huge chunks at a time.
We know that, even though they will heal from serious injury, there are ways to temporarily subdue an original. Their cursed daggers, for example, will put them into a permanent state of suspended animation, and we've seen other magic items take them down as well. We have to assume, if all he had to do was subdue one and then decapitate or cut them up, he would have done so (or at least made an effort to do so) instead of focusing all his effort on acquiring one of the magic weapons.