Video streaming is growing in popularity and has become the most bandwidth-consuming Internet service. As such, robust streaming in terms of low latency and uninterrupted streaming experience, particularly for viewers in distant areas, has become a challenge. The common practice to reduce latency is to pre-process multiple versions of each video and use Content Delivery Networks (CDN) to cache videos that are popular in a geographical area. However, with the fast-growing video repository sizes, caching video contents in multiple versions on each CDN is becoming inefficient. Accordingly, in this paper, we propose the architecture for Fog Delivery Networks (FDN) and provide methods to federate them (called F-FDN) to reduce video streaming latency. In addition to caching, FDNs have the ability to process videos in an on-demand manner. F-FDN leverages cached contents on the neighboring FDNs to further reduce latency. In particular, F-FDN is equipped with methods that aim at reducing latency through probabilistically evaluating the cost benefit of fetching video segments either from neighboring FDNs or by processing them. Experimental results against alternative streaming methods show that both on-demand processing and leveraging cached video segments on neighboring FDNs can remarkably reduce streaming latency (on average 52%).