Synchronous federated learning (FL) is a popular paradigm for collaborative edge learning. It typically involves a set of heterogeneous devices locally training neural network (NN) models in parallel with periodic centralized aggregations. As some of the devices may have limited computational resources and varying availability, FL latency is highly sensitive to stragglers. Conventional approaches discard incomplete intra-model updates done by stragglers, alter the amount of local workload and architecture, or resort to asynchronous settings; which all affect the trained model performance under tight training latency constraints. In this work, we propose straggler-aware layer-wise federated learning (SALF) that leverages the optimization procedure of NNs via backpropagation to update the global model in a layer-wise fashion. SALF allows stragglers to synchronously convey partial gradients, having each layer of the global model be updated independently with a different contributing set of users. We provide a theoretical analysis, establishing convergence guarantees for the global model under mild assumptions on the distribution of the participating devices, revealing that SALF converges at the same asymptotic rate as FL with no timing limitations. This insight is matched with empirical observations, demonstrating the performance gains of SALF compared to alternative mechanisms mitigating the device heterogeneity gap in FL.