The effect of strain due to lattice mismatch and of ferromagnetic exchange field on superconductivity (SC) in NbN-CoPt bilayers is investigated. Two different bilayer systems with reversed deposition sequence are grown on MgO (001) single crystals. While robust superconductivity with high critical temperature $({T}_{c}\ensuremath{\approx}15.3\phantom{\rule{0.3em}{0ex}}\mathrm{K})$ and narrow transition width $(\ensuremath{\Delta}{T}_{c}\ensuremath{\approx}0.4\phantom{\rule{0.3em}{0ex}}\mathrm{K})$ is seen in two types of $\mathrm{Co}\mathrm{Pt}\text{\ensuremath{-}}\mathrm{Nb}\mathrm{N}∕\mathrm{Mg}\mathrm{O}$ heterostructures where the magnetic anisotropy of CoPt is in plane in one case and out of plane in the other, the $\mathrm{Nb}\mathrm{N}\text{\ensuremath{-}}\mathrm{Co}\mathrm{Pt}∕\mathrm{Mg}\mathrm{O}$ system shows markedly suppressed SC response. The reduced SC order parameter of this system, which manifests itself in ${T}_{c}$, temperature dependence of critical current density ${J}_{c}(T)$, and angular $(\ensuremath{\phi})$ variation of flux-flow resistivity ${\ensuremath{\rho}}_{f}$, is shown to be a signature of the structure of NbN film and not a result of the exchange field of CoPt. The ${\ensuremath{\rho}}_{f}$ $(H,T,\ensuremath{\phi})$ data further suggest that the domain walls in the CoPt film are of the N\'eel type and hence do not cause any flux in the superconducting layer. A small but distinct increase in the low-field critical current of the CoPt-NbN couple is seen when the magnetic layer has perpendicular anisotropy.