Emergent $s+id$ Superconductivity from the Interplay between Electronic Correlations and Electron-Phonon Coupling in ${\mathrm{R}}_{1-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{N}\mathrm{i}\mathrm{O}}_2$

Recent tunneling measurements on infinite-layer nickelates have revealed spatially varying superconducting symmetries, whose microscopic origin remains unclear. Motivated by this observation, we investigate the interplay between electron correlations and electron-phonon interactions in infinite-layer nickelates by combining first-principles calculations with the fluctuation-exchange-Migdal-Eliashberg theory. Our calculations show that spin fluctuations yield robust $d$-wave superconductivity on the Ni $d_{x^2-y^2}$ orbital, whereas electron-phonon coupling induces $s$-wave pairing on an interstitial orbital, leading to an $s+id$ superconducting state. The emergence of the $s$-wave component is strongly carrier-density dependent: an intermediate electron-phonon coupling of $\lambda =0.4$ stabilizes the $s+id$ state at $n=0.9$ but not at $n=0.8$. These results imply that local oxygen defects tune the local electron density and form finite-size domains with distinct pairing symmetries, offering a compelling explanation for the spatially inhomogeneous superconducting symmetries observed in experiments.