The rise of Federated Learning (FL) is bringing machine learning to edge computing by utilizing data scattered across edge devices. However, the heterogeneity of edge network topologies and the uncertainty of wireless transmission are two major obstructions of FL's wide application in edge computing, leading to prohibitive convergence time and high communication cost. In this work, we propose an FL scheme to address both challenges simultaneously. Specifically, we enable edge devices to learn self-distilled neural networks that are readily prunable to arbitrary sizes, which capture the knowledge of the learning domain in a nested and progressive manner. Not only does our approach tackle system heterogeneity by serving edge devices with varying model architectures, but it also alleviates the issue of connection uncertainty by allowing transmitting part of the model parameters under faulty network connections, without wasting the contributing knowledge of the transmitted parameters. Extensive empirical studies show that under system heterogeneity and network instability, our approach demonstrates significant resilience and higher communication efficiency compared to the state-of-the-art.