Unmanned aerial vehicles (UAVs), when interconnected in a multi-hop ad-hoc fashion, or as a flying ad-hoc network (FANET), can efficiently accomplish mission-critical tasks. However, UAVs usually suffer from the issues of shorter lifespan and limited computational resources. Therefore, the existing security approaches, being fragile, are not capable of countering the attacks, whether known or unknown. Such a security lapse can result in a debilitated FANET system. In order to cope up with such attacks, various efficient signature schemes have been proposed. Unfortunately, none of the solutions work effectively because of incurred computational and communication costs. We aimed to resolve such issues by proposing a blind signature scheme in a certificateless setting. The scheme does not require public-key certificates, nor does it suffer from the key escrow problem. Moreover, the data that are aggregated from the platform that monitors the UAVs might be too huge to be processed by the same UAVs engaged in the monitoring task. Due to being latency-sensitive, it demands high computational capability. Luckily, the envisioned fifth generation (5G) mobile communication introduces multi-access edge computing (MEC) in its architecture. MEC, when incorporated in a UAV environment, in our proposed model, divides the workload between UAVs and the on-board microcomputer. Thus, our proposed model extends FANET to the 5G mobile network and enables a secure communication between UAVs and the base station (BS).