We investigate a pre-stressed annular thin film subjected to a uniform displacement field along its inner boundary. This loading scenario leads to a variable stress distribution characterized by an orthoradial component that may change sign along a concentric circle within the annular domain. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a localized wrinkling pattern near the inner edge. Using a linear non-homogeneous pre-bifurcation state, the eigenvalue problem describing this instability is cast as a singularly-perturbed fourth-order linear differential equation with variable coefficients. The dependence of the lowest eigenvalue on various non-dimensional quantities is numerically investigated using the compound matrix method. These results are complemented by a WKB analysis which suggests that the qualitative and quantitative features of the full model can be described by a simplified second-order eigenvalue problem which takes into account the finite stiffness of the system.