The development of the sixth generation (6G) of wireless networks is bound to streamline the transition of computation and learning towards the edge of the network. Hierarchical federated learning (HFL) becomes, therefore, a key paradigm to distribute learning across edge devices to reach global intelligence. In HFL, each edge device trains a local model using its respective data and transmits the updated model parameters to an edge server for local aggregation. The edge server, then, transmits the locally aggregated parameters to a central server for global model aggregation. The unreliability of communication channels at the edge and backhaul links, however, remains a bottleneck in assessing the true benefit of HFL-empowered systems. To this end, this paper proposes an unbiased HFL algorithm for unmanned aerial vehicle (UAV)-assisted wireless networks that counteracts the impact of unreliable channels by adjusting the update weights during local and global aggregations at UAVs and terrestrial base stations (BS), respectively. To best characterize the unreliability of the channels involved in HFL, we adopt tools from stochastic geometry to determine the success probabilities of the local and global model parameter transmissions. Accounting for such metrics in the proposed HFL algorithm aims at removing the bias towards devices with better channel conditions in the context of the considered UAV-assisted network.. The paper further examines the theoretical convergence guarantee of the proposed unbiased UAV-assisted HFL algorithm under adverse channel conditions. One of the developed approach's additional benefits is that it allows for optimizing and designing the system parameters, e.g., the number of UAVs and their corresponding heights. The paper results particularly highlight the effectiveness of the proposed unbiased HFL scheme as compared to conventional FL and HFL algorithms.