Two types of domain walls exist in magnetically soft cylindrical nanowires: the transverse-vortex wall (TVW) and the Bloch-point wall (BPW). The latter is expected to prevent the usual Walker breakdown, and thus enable high domain wall speed. We showed recently [M. Sch\"obitz \etal, Phys. Rev. Lett. 123, 217201 (2019)] that the previously overlooked OErsted field associated with an electric current is a key in experiments to stabilize the BPW and reach speed above 600 m/s with spin-transfer. Here, we investigate in detail this situation with micromagnetic simulations and modeling. The switching of the azimuthal circulation of the BPW to match that of the OErsted field occurs above a threshold current scaling with $1/R^3$ ($R$ is the wire radius), through mechanisms that may involve the nucleation and/or annihilation of Bloch points. The domain wall dynamics then remains of a below-Walker type, with speed largely determined by spin-transfer torque alone.
