Drone-assisted radio communication is revolutionizing future wireless networks, including sixth-generation (6G) and beyond, by providing unobstructed, line-of-sight links from air to terrestrial vehicles, enabling robust cellular cehicle-to-everything (C-V2X) communication networks. However, addressing communication latency is imperative, especially when considering autonomous vehicles. In this study, we analyze different types of delay and the factors impacting them in drone-assisted C-V2X networks. We specifically investigate C-V2X Mode 4, where multiple vehicles utilize available transmission windows to communicate the frequently collected sensor data with an embedded drone server. Through a discrete-time Markov model, we assess the medium access control (MAC) layer performance, analyzing the trade-off between data rates and communication latency. Furthermore, we compare the delay between cooperative perception messages (CPMs) and periodically transmitted basic safety messages (BSMs). Our simulation results emphasize the significance of optimizing BSM and CPM transmission intervals to achieve lower average delay as well as utilization of drones’ battery power to serve the maximum number of vehicles in a transmission time interval (TTI). The results also reveal that the average delay heavily depends on the packet arrival rate while the processing delay varies with the drone occupancy and state-transition rates for both BSM and CPM packets. Furthermore, an optimal policy approximates a threshold-based policy in which the threshold depends on the drone utilization and energy availability.