For Mantle discussion, please read this (viewtopic.php?f=16&t=25420).
AFAIK, DirectX was considered being used in the past but that didn't work out. Thus, I don't have any reason to believe that a new DirectX version will be used by F@H.
ETA:
Now ↞ Very Soon ↔ Soon ↔ Soon-ish ↔ Not Soon ↠ End Of Time
Generally speaking, competition is a good thing. If DX and Mantle push AMD and MS to improve their respective product, everyone benefits. If they agree on a global standard and then out-compete with each other in producing faster/better adherence to standards, even better.
Personally, my interest lies in the compute APIs like CUDA, OpenCL, and DirectCompute. Not surprisingly, both Microsoft and nVidia have chosen to deliver a superior proprietary solution while providing an inferior OpenCL compute API for their GPUs. (ATI/AMD has provided a superior version of OpenCL.)
Of course it's difficult to compare one with the other because in the final analysis, we see the combined performance of the driver software and the hardware, itself.