  In this paper, we derive the consistent thermodynamics of the
four-dimensional Lorentzian charged Reissner-Nordstr\"om-NUT (RN-NUT),
Kerr-Newman-NUT (KN-NUT), and RN-NUT-AdS spacetimes in the framework of the
($\psi-\mathcal{N}$)-pair formalism, and then investigate their topological
numbers by using the uniformly modified form of the generalized off-shell
Helmholtz free energy. We find that these solutions can be included into one of
three categories of those well-known black hole solutions, which implies that
these spacetimes should be viewed as generic black holes from the perspective
of the topological thermodynamic defects. In addition, we demonstrate that
although the existence of the NUT charge parameter seems to have no impact on
the topological number of the charged asymptotically locally flat spacetimes,
it has a remarkable effect on the topological number of the charged
asymptotically locally AdS spacetime.
