Heterogeneous aerial access networks (AANs), comprising a hierarchical model of a low Earth orbit (LEO) satellite constellation and multiple high-altitude platforms (HAPs), provide a radio access medium from the sky to enhance the service experience of ground terminals (GTs) in remote areas. In this scenario, aerial edge caching addresses the challenge of delivering on-demand content requests to GTs. Although existing caching schemes for AANs have improved caching efficiency, they often overlook important concerns related to privacy preservation and overhead. This paper proposes an intelligent, proactive content-caching scheme to maximize cache efficiency while ensuring privacy preservation for GTs. The proposed caching scheme employs hierarchical federated learning (HierFL) that involves GTs, HAPs, and LEO satellites in heterogeneous AANs. In particular, the proposed HierFL-based proactive content-caching (HierFL-PCC) scheme leverages a deep neural network employing multiple linear regression to predict the dynamically changing content popularity of GTs, maximizing the cache efficiency. The HierFL-PCC also reduces the delay and overhead associated with content delivery. Additionally, the proposed HierFL-PCC scheme reduces communication overhead in model training owing to the hierarchical learning architecture. Simulation results demonstrate that the proposed HierFL-PCC scheme exhibits improvement of about 13%, 43%, 178%, and 457% in cache efficiency, and reduces average content delivery delay by approximately 10%, 20%, 31%, and 37%, along with an improvement of around 89% in training overhead, respectively, compared to the cloud-based FAVG, LFU, LRU, and Random caching schemes.