Ensuring dependable quality of service (QoS) and quality of experience (QoE) for computation-intensive and delay-sensitive applications in vehicles can be a challenging task that impacts performance. While multi-access edge computing (MEC) based vehicular edge computing network (VECN) and vehicular cloudlets (VC) enable task offloading, but their prompt and optimal accessibility is another challenge. The conventional wireless technologies may not suffice to meet the stringent ultra-low latency and cost constraints of such applications. Nonetheless, the combination of different wireless technologies can enhance network performance and satisfy these requirements. Focusing on the computational efficacy of VECN, this paper proposes a  mobility, contact, and computational load-aware (MCLA) task offloading scheme for heterogeneous VECN. The MCLA scheme dynamically considers the mobility, contact, and computational load of vehicles for making task offloading decisions. To optimize the performance, the MCLA scheme integrates the Mode-1 and Mode-2 of the 5G-NR-V2X standard, along with mmWave communications. The MCLA scheme provides an opportunistic switching mechanism between these modes and heterogeneous radio access technologies (RATs) to reduce communication delays and costs. Moreover, the MCLA scheme leverages public vehicles (i.e., public buses), in proximity by using their computational power to manage computational latency and cost. Furthermore, it also considers the shareable computations from passengers’ mobile equipment within the public vehicle to improve the computation capacity of the public vehicles. Extensive evaluations and numerical results show that the proposed MCLA scheme significantly improves the task turnover ratio by 4%-15% with 4.7%-29.8% lower transmission and computation costs.