For every non-vanishing spinor field on a Riemannian $7$-manifold, Crowley, Goette, and Nordström introduced the so-called nu-invariant. This is an integer modulo $48$, and can be defined in terms of Mathai-Quillen currents, harmonic spinors, and eta-invariants of spin Dirac and odd-signature operator. In general, the role of the nu-invariant is to detect connected components of the moduli space of $G_2$-structures. We explain how we compute these data on compact two-step nilmanifolds admitting invariant closed $G_2$-structures, in particular determining harmonic spinors and relevant symmetries of the spectrum of the spin Dirac operator.