CdSe/CdTe core-crown type-II nanoplatelet heterostructures are two-dimensional semiconductors that have attracted interest for use in light-emitting technologies due to their ease of fabrication, outstanding emission yields, and tunable properties. Despite this, the exciton dynamics of these complex materials, and in particular how they are influenced by phonons, is not yet well understood. Here, we use a combination of femtosecond vibrational spectroscopy, temperature-resolved photoluminescence (PL), and temperature-dependent structural measurements to investigate CdSe/CdTe nanoplatelets with a thickness of four monolayers. We show that charge-transfer (CT) excitons across the CdSe/CdTe interface are formed on two distinct time scales: initially from an ultrafast (∼70 fs) electron transfer and then on longer time scales (∼5 ps) from the diffusion of domain excitons to the interface. We find that the CT excitons are influenced by an interfacial phonon mode at ∼120 cm-1, which localizes them to the interface. Using low-temperature PL spectroscopy we reveal that this same phonon mode is the dominant mechanism in broadening the CT PL. On cooling to 4 K, the total PL quantum yield reaches close to unity, with an ∼85% contribution from CT emission and the remainder from an emissive sub-band-gap state. At room temperature, incomplete diffusion of domain excitons to the interface and scattering between CT excitons and phonons limit the PL quantum yield to ∼50%. Our results provide a detailed picture of the nature of exciton-phonon interactions at the interfaces of 2D heterostructures and explain both the broad shape of the CT PL spectrum and the origin of PL quantum yield losses. Furthermore, they suggest that to maximize the PL quantum yield both improved engineering of the interfacial crystal structure and diffusion of domain excitons to the interface, e.g., by altering the relative core/crown size, are required.