Federated learning (FL) is capable of performing large distributed machine learning tasks across multiple edge users by periodically aggregating trained local parameters. To address key challenges of enabling FL over a wireless fog-cloud system (e.g., non-i.i.d. data, users’ heterogeneity), we first propose an efficient FL algorithm based on Federated Averaging (called FedFog) to perform the local aggregation of gradient parameters at fog servers and global training update at the cloud. Next, we employ FedFog in wireless fog-cloud systems by investigating a novel network-aware FL optimization problem that strikes the balance between the global loss and completion time. An iterative algorithm is then developed to obtain a precise measurement of the system performance, which helps design an efficient stopping criteria to output an appropriate number of global rounds. To mitigate the straggler effect, we propose a flexible user aggregation strategy that trains fast users first to obtain a certain level of accuracy before allowing slow users to join the global training updates. Extensive numerical results using several real-world FL tasks are provided to verify the theoretical convergence of FedFog. We also show that the proposed co-design of FL and communication is essential to substantially improve resource utilization while achieving comparable accuracy of the learning model.