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When implementing hierarchical federated learning over wireless networks, scalability assurance and the ability to handle both interference and device data heterogeneity are crucial. This work introduces a new two-level learning method designed to address these challenges, along with a scalable over-the-air aggregation scheme for the uplink and a bandwidth-limited broadcast scheme for the downlink that efficiently use a single wireless resource. To provide resistance against data heterogeneity, we employ gradient aggregations. Meanwhile, the impact of uplink and downlink interference is minimized through optimized receiver normalizing factors. We present a comprehensive mathematical approach to derive the convergence bound for the proposed algorithm, applicable to a multi-cluster wireless network encompassing any count of collaborating clusters, and provide special cases and design remarks. As a …