This work studies the joint design of cloud and edge processing for latency minimization in the downlink of a fog radio access network (F-RAN). In an F-RAN, edge processing allows the low-latency delivery of popular multimedia content by leveraging caching at enhanced remote radio heads (eRRHs). Cloud processing, instead, enables the transmission of arbitrary content at high spectral efficiencies thanks to the centralized control at a baseband processing unit (BBU), but at the cost of a potentially larger latency due to BBU-to-eRRHs communication over fronthaul links. For an arbitrary caching, or pre-fetching, strategy, the design of the delivery phase is studied based on the use of the fronthaul links in either or both hard-transfer and soft-transfer modes. The problem of minimizing the delivery latency, encompassing both fronthaul and wireless transmissions, of the requested contents from the BBU to the requesting user equipments (UEs) is tackled with respect to channel precoding and fronthaul compression strategies. Numerical results are provided to compare the latency performance of the hard- and soft-transfer fronthauling schemes in terms of delivery latency, offering new insights as compared to existing studies that focus solely on the transmission rate of the wireless segment.