Is it possible to redistribute NHRP learned routes (H) into another protocol and vice versa? If yes, what does the syntax look like?
I'm not aware of this as being possible and I don't think it is because as NHRP routes appear in dmvpn phase 3, they are bassed on the underlying protocol that's being deployed between hub and spokes.
NHRP is not a routing protocol. It is used to resolve next hop addresses over a NBMA network. In some respects it's closer to ARP than a routing protocol. It is not possible to redistribute to/from NHRP.
Maybe you could explain what you are trying to achieve?
Can you not just redistribute the overlay protocol (ospf in your case) - the next hop on the DMVPN router will get sorted out on the fly with NHRP and a dynamic tunnel - and the route you send downstream/upstream will get the DMVPN router as the next hop, so there should not be a problem.
As NHRP installs routes in the RIB, behind the scenes it is a routing protocol, but it is not configurable at this point (it's intrinsec behaviour is built-in the DMVPN code). With SDN it may become available for further development.
even though NHRP is not limited to DMVPN....it did exist before DMVPN from my understanding...