People are thinking that DX12 is going to replace Mantle. DX12 is not in a good position. It is Windows 10 only and Microsoft is releasing DX 11.3 to compete with DX12 and only to leave DX12 for developers who want lower level options.
In the future, you're going to see Mantle still be used by the types of developers who are more inclined to do technical stuff and push boundaries. DX11.3 will be used by developers who are opting for things like a AAA title with DLC cash grab.
I'm also assuming that Mantle will continue to serve as a beta testing ground for AMD and their partners to commit code to Vulkan. Companies like Oxide aren't going to abandon Mantle, they are doing great things with it and they have no governing body to try and petition to get changes made. I don't think Mantle will ever be a mainstream thing, but it will be the source of some cool tech demos and some good innovations to Vulkan.
Valve is rumored to be doing a lot of stuff with Vulkan, we'll have to see. But it sounds like, if FX 8350 is saying is true, that AMD just handed over large parts of Mantle to Khronos and Valve so Valve could take care of porting it to other platforms (like Linux/SteamOS) and they wouldn't have to deal with that. Mantle is designed to push hardware in ways previous APIs couldn't. AMD simply doesn't have the resources (even though they say they do) to waste maintaining an API on Linux, Windows, Android, etc. like Vulkan will support. They are supposed to be giving a good talk on Vulkan today, actually. There's also something with Oxide and AMD with Mantle that's supposed to be announced.
It's too early to completely write off Mantle. The developers who would rather create "cinematic experiences" and "some safe-market game with loads of DLC" will all abandon Mantle for DX12 and Vulkan. But the developers like Oxide, who like to push tech, are going to stay with Mantle.
The biggest winner out of all of this seems to be Vulkan. It's going to be far more open than Mantle or DX12, and it's not going to be limited to a specific OS or specific hardware. I'm assuming most people are going to take Vulkan on whatever OS they want (and this includes Windows 7!) as opposed to jumping into Windows 10.
Vulkan is going to be interesting for Nvidia. It relies heavily on OpenCL and Nvidia has traditionally "not optimized" (or actively harmed) OpenCL performance to encourage developers to stay with CUDA.
http://www.phoronix.com/scan.php?page=article&item=khronos-vulcan-spirv&num=1
The most important thing though, is that things are changing. We're going to get a lot of new, exciting things at a time when graphics have grown stale and not improving so much, as evidenced by 3 year old graphic cards being more than enough and people being reduced to arguing about performance on 4k and 8k resolutions because the common gaming resolutions are completely covered by mid range cards.
Also, I called this long ago, and I'm going to call it now. The future of graphic APIs will be Nvidia pushing Windows 10 and DX12 against the entire world. Nvidia has nothing to gain from Mantle and it looks like all their sabotaging OpenCL performance over all these years is going to really hurt their Vulkan performance since Vulkan relies so heavily on OpenCL.
http://developer.amd.com/community/events/amd-gdc-2015/
It looks like the big Oxide Games tech demo that I've seen kicked around is on March 5th.