This paper delves into the time-efficient covert multicast in a wireless communication system facilitated by Unmanned Aerial Vehicle (UAV), in which the UAV aims to disseminate a common covert information to multiple ground users (GUs) while suffering from the risk of detection by a ground warden (Willie). We propose one hop (OH) and two hop (TH) transmission schemes, first develop a theoretical framework for performance modeling of both the detection error probability at Willie and the transmission time at UAV. The optimization problems subject to the covertness constraint for the two transmission schemes are then formulated to gain insights into the system settings of the UAV's prior transmit probability, transmit power and horizontal location that affect the minimum transmission time. The optimization problems are non-convex and challenging to give numerical results. We thus explore the optimal setting of the transmit power and the prior transmit probability for the UAV separately under specific parameters with two schemes. We further propose a particle swarm optimization (PSO) based algorithm and an exhaustive algorithm to provide the joint solutions for the optimization problem with the OH transmission scheme and TH scheme, respectively. Finally, the efficiency of the proposed PSO-based algorithm is substantiated through extensive numerical results.