The convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is investigated via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability. This oscillator-type flow behaviour is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in part 1 of this study (d'Olce, Martin, Rakotomalala, Salin and Talon, J. Fluid Mech., vol. 618, 2008, pp. 305–322), self-excited synchronized oscillations were also observed experimentally. An interpretation of these experiments is attempted on the basis of the numerical results presented here.