The rapid proliferation of Internet of Things (IoT) ground devices (GDs) has created an unprecedented demand for computing resources and real-time data-processing capabilities. Integrating unmanned aerial vehicles (UAVs) into Mobile Edge Computing (MEC) emerges as a promising solution to bring computation and storage closer to the data sources. However, UAV heterogeneity and the time window constraints for task execution pose a significant challenge. This paper addresses the multiple heterogeneity UAV routing problem in MEC environments, modeling it as a multi-traveling salesman problem (MTSP) with soft time constraints. We propose a two-stage heuristic algorithm, heterogeneous multiple UAV routing (HMUR). The approach first identifies task areas (TAs) and optimal hovering positions for the UAVs and defines an effective fitness measurement to handle UAV heterogeneity. A novel scoring function further refines the path determination, prioritizing real-time task compliance to enhance Quality of Service (QoS). The simulation results demonstrate that our proposed HMUR method surpasses the existing baseline algorithms on multiple metrics, validating its effectiveness in optimizing resource scheduling in MEC environments.