We present a comprehensive study on the post-buckling response of nonlocal structures performed by means of a frame-invariant fractional-order continuum theory to model the long-range (nonlocal) interactions. The use of fractional calculus facilitates an energy-based approach to nonlocal elasticity that plays a fundamental role in the present study. The underlying fractional framework enables mathematically, physically, and thermodynamically consistent integral-type constitutive models that, in contrast to the existing integer-order differential approaches, allow the nonlinear buckling and post-bifurcation analyses of nonlocal structures. Further, we present the first application of the Koiter's asymptotic method to investigate post-bifurcation branches of nonlocal structures. Finally, the theoretical framework is applied to study the post-buckling behavior of slender nonlocal plates. Both qualitative and quantitative analyses of the influence that long-range interactions bear on post-buckling response are undertaken. Numerical studies are carried out using a 2D fractional-order Finite Element Method (f-FEM) modified to include a combination of the Newton-Raphson and a path-following arc-length iterative methods in order to solve the system of nonlinear algebraic equations that govern the equilibrium beyond the critical points. The present framework provides a general foundation to investigate the post-buckling response of potentially any type of nonlocal structure.