The light-mediated interaction of fermionic and bosonic excitations governs the optoelectronic properties of condensed matter systems. In photoexcited semiconductors, the coupling of electron-hole pairs (excitons) to coherent optical phonons enables a modulation of the excitonic resonance that is phase-locked to the frequency of the coupled vibrational mode. Moreover, due to the Coulombic nature of excitons, their dynamics are sensitive to transient changes in the screening by the photoexcited carriers. Interestingly, the effect of photoinduced screening on the transient optical signal originating from the exciton dynamics coupled to phonons is not yet established. By means of broadband transient reflectance spectroscopy, we disclose how exciton-phonon coupling manifests in either the presence or absence of dynamical screening in a layered semiconductor. Furthermore, we unveil the promoting role of photoinduced screening on these exciton-phonon coupled dynamics as opposed to the case in which the unscreened exciton-exciton repulsion likely dominates the nonequilibrium optical response. These findings set a protocol to look at an excitonic resonance and its fundamental many-body interactions on the ultrafast timescale and provide new perspectives on the access to nonequilibrium coupled dynamics.