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. Schöbitz, Phys. Rev. Lett. 123, 217201 (2019)10.1103/PhysRevLett.123.217201] that the previously overlooked Oersted field associated with an electric current is a key in experiments to stabilize the BPW and reach speed above 600m/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/R3 (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.