T Tauri stars produce broad Lyα emission lines that contribute ~88% of the total UV flux incident on the inner circumstellar disks. Lyα photons are generated at the accretion shocks and in the protostellar chromospheres and must travel through accretion flows, winds, and jets, the protoplanetary disks, and the interstellar medium before reaching the observer. This trajectory produces asymmetric, double-peaked features that carry kinematic and opacity signatures of the disk environments. To understand the link between the evolution of Lyα emission lines and the disks themselves, we model HST-COS spectra from targets included in Data Release 3 of the Hubble UV Legacy Library of Young Stars as Essential Standards program. We find that resonant scattering in a simple spherical expanding shell is able to reproduce the high-velocity emission line wings, providing estimates of the average velocities within the bulk intervening H I. The model velocities are significantly correlated with the K-band veiling, indicating a turnover from Lyα profiles absorbed by outflowing winds to emission lines suppressed by accretion flows as the hot inner disk is depleted. Just 30% of targets in our sample have profiles with redshifted absorption from accretion flows, many of which have resolved dust gaps. At this stage, Lyα photons may no longer intersect with disk winds along the path to the observer. Our results point to a significant evolution of Lyα irradiation within the gas disks over time, which may lead to chemical differences that are observable with ALMA and JWST.