This seems to be the opposite of "Mantle under a new name" type arguments...
AMD must have known Nvidia wouldn't support Mantle when they first thought it up. Most of us could have figured that out. It would've been stupid if Nvidia did. I can't believe AMD would've if the roles were reversed. So AMD developed Mantle knowing full well that it wasn't going to be completely cross-vendor (how Intel would react is sort of irrelevant for a gaming API at the current time).
Nvidia supporting a 3rd party controlled API is very different from supporting one controlled by your big rival. Nvidia may have known about DX12 by the time Mantle was being publicly talked about and that may have been their reason for not seeing the point in supporting it. Vulkan, as mentioned is different and Nvidia have had a lot to do with OpenGL in the past I believe, so no reason they wouldn't support its successor.
So, was Mantle responsible for Vulkan, I think it did play a large part, but I think there was more work to be done than renaming it.
But Mantle, like GameWorks wasn't this perfect piece of software that was open source for both vendors. However the position we're now in with Vulkan and DX12 can only be a good thing. So I think a rival to GameWorks would be a good thing, to keep both moving forwards. Having one unified library might stop innovation.
Still not entirely sure how Mantle/Vulkan fits into a GameWorks thread though...