In this paper, we consider a finite network of unmanned aerial vehicles (UAVs) serving a given region. Modeling this network as a uniform binomial point process (BPP), we derive the downlink coverage probability of a reference receiver located at an arbitrary position on the ground assuming Nakagami-$m$ fading for all wireless links. The reference receiver is assumed to connect to its closest transmitting node as is usually the case in cellular systems. After deriving the distribution of distances from the reference receiver to the serving and interfering nodes, we derive an exact expression for downlink coverage probability in terms of the derivative of Laplace transform of interference power distribution. In the downlink of this system, it is not unusual to encounter scenarios in which the line-of-sight (LOS) component is significantly stronger than the reflected multipath components. To emulate such scenarios, we also derive the coverage probability in the absence of fading from the results of Nakagami-$m$ fading by taking the limit $m \to \infty$. Using asymptotic expansion of incomplete gamma function, we concretely show that this limit reduces to a redundant condition. Consequently, we derive an accurate coverage probability approximation for this case using dominant interferer-based approach in which the effect of dominant interferer is exactly captured and the residual interference from other interferers is carefully approximated. We then derive the bounds of the approximate coverage probability using Berry-Esseen theorem. Our analyses reveal several useful trends in coverage probability as a function of height of the transmitting nodes and the location of reference receiver on the ground.