In this paper, we will study the phenomenon of neutrino oscillations when traveling through the gravitational field of a black hole embedded in a spatially homogeneous and isotropic flat universe. Thus, we will compute the phase shift for neutrinos within the McVittie spacetime building upon the description of their geodesic motion. Standard assumptions of ultrarelativistic neutrinos with degeneracy between their energy eigenstates are maintained, while no screening mechanisms are needed; the correctness of our result can therefore be checked against relevant literature limiting scenarios of static isolated Schwarzschild black hole and of the Friedmann universe backgrounds, separately. We will compare and contrast the effects of local galactic and global cosmic physics, accounted for by the black hole mass and the Hubble rate, in the phase shift; its numerical value will be estimated for some realistic scenarios by re-casting it in terms of measurable astrophysical and cosmological parameters. We will identify the range of applicability of our model as for neutrinos sourced in the accelerated expansion phase of the universe, with the roles of the cosmological constant, dark matter and black hole mass scrutinized in the loss of coherence of the wavepackets. Our analysis will allow us to identify similarities and differences in the interaction between neutrinos and matter or gravitational fields.