In this study, we analyze the capillary filling dynamics of a viscoelastic fluid through a concentric annulus, which has far reaching consequences in practical applications and offers a distinct disparity in the dynamical characteristics as compared to the classical cylindrical capillary based paradigm. Such non-trivial characteristics are primarily attributed to a complex and intricate interplay between the intrinsic fluid rheology and the annular flow geometry, as is effectively manifested through distinctive features of the underlying oscillatory dynamics. We also estimate a criterion for the onset of oscillations, as a function of the Bond number. Our results predict remarkably attenuated oscillatory behavior and a higher capillary rise due to the presence of an annular geometry, as compared to a cylindrical one. We further relate the primary peak overshoot response with the Bond number that enables us to draw further physical insights into the oscillatory regime dynamics.