Connected autonomous vehicles (CAVs), through vehicle-to-everything communication and computing resources, enable the vital exchange of information. Although deep learning is crucial in this landscape, it requires extensive and intricate datasets covering all potential scenarios. Furthermore, this situation poses a hazard, as the likelihood of accidents associated with imbalanced datasets increases, particularly in scenarios where processing analysis is compromised due to fluctuating weather conditions. We propose a Federated Learning (FL) framework undergirded by Multi-Access Edge Computing (MEC) to counter these challenges. This local device-focused framework enhances task-specific models' caching and continual updating across various conditions. In a more specific sense, edge nodes (ENs) operate as MEC, each caching multiple dedicated models and serving as the aggregator as part of the FL process. Additionally, we have engineered two innovative algorithms that categorize various states into multiple classes, thereby ensuring the efficient utilization of computing resources in ENs. Simulation results substantiate the effectiveness of our approach, showing that the proposed dedicated model consistently outperforms a general model designed for all situations.