The futuristic sixth-generation (6G) networks will empower ultra-reliable and low latency communications (URLLC), enabling a wide array of mission-critical applications such as mobile edge computing (MEC) systems, which are largely unsupported by fixed communication infrastructure. To remedy this issue, unmanned aerial vehicle (UAV) has recently come to the limelight to facilitate MEC for internet of things (IoT) devices as they provide desirable line-of-sight (LoS) communications compared to fixed terrestrial networks, thanks to their added flexibility and three-dimensional (3D) positioning. In this paper, we consider UAV-enabled relaying for MEC systems for uplink transmissions in 6G networks, and we aim to optimize mission completion time subject to the constraints of resource allocation, including UAV transmit power, UAV CPU frequency, decoding error rate, blocklength, communication bandwidth, and task partitioning as well as 3D UAV positioning. Moreover, to solve the non-convex optimization problem, we propose three different algorithms, including successive convex approximations (SCA), altered genetic algorithm (AGA) and smart exhaustive search (SES). Thereafter, based on time-complexity, execution time, and convergence analysis, we select AGA to solve the given optimization problem. Simulation results demonstrate that the proposed algorithm can successfully minimize the mission completion time, perform power allocation at the UAV side to mitigate information leakage and eavesdropping as well as map a 3D UAV positioning, yielding better results compared to the fixed benchmark sub-methods. Lastly, subject to 3D UAV positioning, AGA can also effectively reduce the decoding error rate for supporting URLLC services.