We computed spin-wave dispersions of surface-reconstructed Co films on the W(110) surface in the adiabatic approximation. The magnetic exchange interactions are obtained via first-principles electronic-structure calculations using the Korringa-Kohn-Rostoker Green-function method. We analyze the strength and oscillatory behavior of the intralayer and interlayer magnetic interactions and investigate the resulting spin-wave dispersions as a function of the thickness of Co films. In particular, we highlight and explain the strong impact of hybridization of the electronic states at the Co-W interface on the magnetic exchange interactions and on the spin-wave dispersions. We compare our results to recent measurements based on electron-energy-loss spectroscopy [E. Michel, H. Ibach, and C.M. Schneider, Phys. Rev. B 92, 024407 (2015)]. Good overall agreement with experimental findings can be obtained by considering the possible overestimation of the spin splitting, stemming from the local-spin-density approximation, and adopting an appropriate correction.