This work is focused on the numerical investigation of spin waves that propagate in nonlinear ferromagnet/superconductor bilayered films and periodic structures. The nonlinearity in these hybrid structures emerges due to the non-monotonous dependence of magnetization of a superconducting subsystem on the magnetic field, which is characterized by the superconducting critical field. It is shown that at relatively high amplitudes of spin waves in comparison to the superconducting critical field, the spin-wave spectrum changes drastically: the spin-wave spectral line can either bifurcate or stretch continuously depending on the type of considered superconductor. In addition, in the case of propagation of spin waves with relatively high amplitude in periodic magnonic metamaterials, additional zero-group-velocity modes appear that are known as flatbands. Overall, these findings suggest a versatile way for tunability of the spin-wave spectrum in nonlinear ferromagnet/superconductor structures by changing the excitation signal in respect to the superconducting critical field.