Over the past few years, Fog Radio Access Networks (F-RANs) have become a promising paradigm to support the tremendously increasing demands of multimedia services, by pushing computation and storage functionalities towards the edge of networks, closer to users. In F-RANs, distributed edge caching among Fog Access Points (F-APs) can effectively reduce network traffic and service latency as it places popular contents at local caches of F-APs rather than the remote cloud. Due to the limited caching resources of F-APs and spatio-temporally fluctuant content demands from users, many cooperative caching schemes were designed to decide which contents are popular and how to cache them. However, these approaches often collect and analyse the data from Internet-of-Things (IoT) devices at a central server to predict the content popularity for caching, which raises serious privacy issues. To tackle this challenge, we propose a Federated Learning based Cooperative Hierarchical Caching scheme (FLCH), which keeps data locally and employs IoT devices to train a shared learning model for content popularity prediction. FLCH exploits horizontal cooperation between neighbour F-APs and vertical cooperation between the BaseBand Unit (BBU) pool and F-APs to cache contents with different degrees of popularity. Moreover, FLCH integrates a differential privacy mechanism to achieve a strict privacy guarantee. Experimental results demonstrate that FLCH outperforms five important baseline schemes in terms of the cache hit ratio, while preserving data privacy. Moreover, the results show the effectiveness of the proposed cooperative hierarchical caching mechanism for FLCH.