Non-terrestrial networks (NTNs) are emerging as a core component of future 6G communication systems, providing global connectivity and supporting data-intensive applications. In this paper, we propose a distributed hierarchical federated learning (HFL) framework within the NTN architecture, leveraging a high altitude platform station (HAPS) constellation as intermediate distributed FL servers. Our framework integrates both low-Earth orbit (LEO) satellites and ground clients in the FL training process while utilizing geostationary orbit (GEO) and medium-Earth orbit (MEO) satellites as relays to exchange FL global models across other HAPS constellations worldwide, enabling seamless, global-scale learning. The proposed framework offers several key benefits: (i) enhanced privacy through the decentralization of the FL mechanism by leveraging the HAPS constellation, (ii) improved model accuracy and reduced training loss while balancing latency, (iii) increased scalability of FL systems through ubiquitous connectivity by utilizing MEO and GEO satellites, and (iv) the ability to use FL data, such as resource utilization metrics, to further optimize the NTN architecture from a network management perspective. A numerical study demonstrates the proposed framework's effectiveness, with improved model accuracy, reduced training loss, and efficient latency management. The article also includes a brief review of FL in NTNs and highlights key challenges and future research directions.