Fog computing is today considered a promising candidate to improve the user experience in dynamic on-demand computing services. However, its ubiquitous application would require support for this service in wireless multi-hop mesh systems, where the use of conventional IP-based solutions is challenging. As a complementary solution, in this paper, we consider a Named-Data Networking (NDN) approach to enable fog computing services in autonomous dynamic mesh formations. In particular, we jointly implement two critical mechanisms required to extend the NDN-based fog computing architecture to wireless mesh systems. These are (i) dynamic face management systems and (ii) a learning-based route discovery strategy. The former makes it possible to solve NDN issues related to an inability to operate over a broadcast medium. Also, it improves the data-link layer reliability by enabling unicast communications between mesh nodes. The learning-based forwarding strategy, on the other hand, efficiently reduces the amount of overhead needed to find routes in the dynamically changing mesh networks. Our numerical results show that, for static wireless meshes, our proposal makes it possible to fully benefit from the computing resources sporadically available up to several hops away from the consumer. Additionally, we investigate the impacts of various traffic types and NDN caching capabilities, revealing that the latter result in much better system performance while the popularity of the compute service contributes to additional performance gains.