Buckled beams are receiving attention as potential candidates for nano- or micro-electromechanical applications such as mechanical sensors, actuators, energy harvesting devices, particularly in buckling-induced smart applications. The mechanical behaviour of such nanostructures is significantly influenced by long-range molecular interactions that can be modelled by gradient-based higher-order continuum theories. The stress on the material surface results in an unconventional elastic response due to high surface-to-bulk ratio of nanostructures. The geometric nonlinearity of the nanostructures, owing to their slenderness, in combination with the molecular interaction forces and surface effects will give a thorough comprehension to the characteristics of these nanostructures. This study investigates dynamic behaviour of a post-buckled nano beam subjecting to large-amplitude nonlinear vibrations. The beam is modelled using the higher order shear deformation theory along with von Kármán nonlinearity and is assumed to be rested on a Pasternak-type substrate. The governing equations of motions are derived considering the nonlocal strain gradient field theory for a laminated magneto-electro-elastic nano beam with the aid of variational principles. The set of nonlinear partial differential equations are solved by employing the two-step perturbation. The closed-form solution is obtained characterising the nonlinear frequency and the response time history of the buckled nano beam. Influence of important parameters such as the effects of the nonlocal and strain-gradient length-scale parameters, the surface stress effects, the effect of electric and magnetic fields and the effect of nonlinear substrate is demonstrated.