Distributed Machine Learning (DML) has emerged as a disruptive technology that enables the execution of Machine Learning (ML) and Deep Learning (DL) algorithms in proximity to data generation, facilitating predictive analytics services in Smart City environments. However, the real-time analysis of data generated by Smart City Edge Devices (EDs) poses significant challenges. Concept drift, where the statistical properties of data streams change over time, leads to degraded prediction performance. Moreover, the reliability of each computing node directly impacts the availability of DML systems, making them vulnerable to node failures. To address these challenges, we propose a resilience framework comprising computationally lightweight maintenance strategies that ensure continuous quality of service and availability in DML applications. We conducted a comprehensive experimental evaluation using real datasets, assessing the effectiveness and efficiency of our resilience maintenance strategies across three different scenarios. Our findings demonstrate the significance and practicality of our framework in sustaining predictive performance in smart city edge learning environments. Specifically, our enhanced model exhibited increased generalizability when confronted with concept drift. Furthermore, we achieved a substantial reduction in the amount of data transmitted over the network during the maintenance of the enhanced models, while balancing the trade-off between the quality of analytics and inter-node data communication cost.