In this work, we propose a distributed rate allocation algorithm that minimizes the average decoding delay for multimedia clients in inter-session network coding systems. We consider a scenario where the users are organized in a mesh network and each user requests the content of one of the available sources. We propose a novel distributed algorithm where network users determine the coding operations and the packet rates to be requested from the parent nodes, such that the decoding delay is minimized for all clients. A rate allocation problem is solved by every user, which seeks the rates that minimize the average decoding delay for its children and for itself. Since this optimization problem is a priori non-convex, we introduce the concept of equivalent packet flows, which permits to estimate the expected number of packets that every user needs to collect for decoding. We then decompose our original rate allocation problem into a set of convex subproblems, which are eventually combined to obtain an effective approximate solution to the delay minimization problem. The results demonstrate that the proposed scheme eliminates the bottlenecks and reduces the decoding delay experienced by users with limited bandwidth resources. We validate the performance of our distributed rate allocation algorithm in different video streaming scenarios using the NS-3 network simulator. We show that our system is able to take benefit of inter-session network coding for simultaneous delivery of video sessions in networks with path diversity.