This paper presents a fixed-time control design for a class of uncertain under-actuated nonlinear systems (UNS) using a non-singular fast terminal sliding mode control (TSMC) with a radial basis function (RBF) based estimator to achieve the fast convergence and robustness against the uncertain disturbances. The generalized mathematical model of the considered class is first reduced into an equivalent regular form, which is more convenient for any control synthesis design. A fast TSMC is designed for the transformed regular form to improve the control performance and annihilate the associated singularity problem of the conventional TSMC scheme. The steering of sliding manifold and system states in fixed-time is ensured through the Lyapunov stability theory. The RBF-based neural networks are used to adaptively estimate the nonlinear drift functions, which are feedbacked to the control input. The theoretical design, analysis and simulations of cart-pendulum and quadcopter systems demonstrate the feasibility and benefits of the regular form transformation and the designed control design. Comparing the proposed control synthesis with the standard literature presents the attractive nature of the proposed method for such a class.