This paper investigates the finite-time cooperative formation control problem for a heterogeneous system consisting of an unmanned ground vehicle (UGV) - the leader and an unmanned aerial vehicle (UAV) - the follower. The UAV system under consideration is subject to modeling uncertainties, external disturbance as well as actuator faults simultaneously, which is associated with aerodynamic and gyroscopic effects, payload mass, and other external forces. First, a backstepping controller is developed to stabilize the leader system to track the desired trajectory. Second, a robust nonsingular fast terminal sliding mode surface is designed for UAV and finite-time position control is achieved using terminal sliding mode technique, which ensures the formation error converges to zero in finite time in the presence of actuator faults and other uncertainties. Furthermore, by combining the radial basis function neural networks (NNs) with adaptive virtual parameter technology, a novel NN-based adaptive nonsingular fast terminal sliding formation controller (NN-ANFTSMFC) is developed. By means of the proposed adaptive control strategy, both uncertainties and actuator faults can be compensated without the prior knowledges of the uncertainty bounds and fault information. By using the proposed control schemes, larger actuator faults can be tolerated while eliminating control chattering. In order to realize fast coordinated formation, the expected position trajectory of UAV is composed of the leader position information and the desired relative distance with UGV, based on local distributed theory, in the three-dimensional space. The tracking and formation controllers are proved to be stable by the Lyapunov theory and the simulation results demonstrate the effectiveness of proposed algorithms.