UAV-based mobile relay communication systems have an important position in future wireless networks, such as Beyond 5G and 6G, which provide ubiquitous network services. Along this line, UAV-based single-hop relay systems have numerous applications in emergency situations, such as data ferry and assisting post-disaster areas. Although there have been numerous works on optimizing trajectories and other communication resources on such systems, there are few studies on delay performance at the granular packet level. In addition to having homogeneous traffic, such a system can be equipped with diverse traffic with different QoS requirements, owing to the emergence of various applications in today’s communication networks. Based on the traffic requirements, packets with high-priority traffic should be served ahead of those with low-priority ones. To this end, we aim to analytically study the average end-to-end packet delay and buffer overflow performance of such a system at the granular packet level. However, packet-level queuing analysis of heterogeneous traffic is not as straightforward as that of homogeneous traffic. While considering different traffic requirements, we establish a priority queuing model for the source and the UAV nodes based on the stochastic process. Then, on the basis of the established queuing model, the average end-to-end packet delay and buffer overflow probability of the system under different trajectories are studied. Through extensive numerical simulations, we verified the accuracy and effectiveness of the proposed analytical model. The results show that the average packet delay and packet drop probability of high-priority packets under all trajectories are significantly better compared to low-priority packets. UAV trajectory has a great impact on network performance, and an appropriate mobile trajectory that takes full use of channel capacity is superior to a static one.