Single-walled carbon nanotubes (SWNTs), when suspended in viscous media in an applied electric field, are caused to align through an induced dipole moment. The consequent introduction of anisotropy into what was a previously isotropic suspension of SWNTs gives rise to induced optical effects such as birefringence and linear dichroism. If the electric field is applied as a transient this alignment may be followed dynamically. SWNTs are an excellent model of a nanoscopic cylinder the dynamics of which is a matter of considerable interest in the emerging field of nanohydrodynamics. In part I of this series of two papers the magnitude of the effect was used to obtain the polarizability of the metallic nanotubes. In this paper the dynamics of SWNT rotation are studied. How the rotation dynamics are affected by the viscosity of different suspending media is compared and contrasted with the predictions of classical hydrodynamic models describing rotation of macro- and mesoscopic objects in viscous media. Such models use continuum hydrodynamic methods where the molecular details of the rotating object and the medium are ignored. The extreme change of scale involved in the case of SWNTs nevertheless does not appear to prevent these classical models being applied, and a local viscosity is established for each medium used that is not greatly different from the accepted macroscopic value.