Neural networks (NNs) have been shown to learn complex control laws successfully, often with performance advantages or decreased computational cost compared to alternative methods. Neural network controllers (NNCs) are, however, highly sensitive to disturbances and uncertainty, meaning that it can be challenging to make satisfactory robustness guarantees for systems with these controllers. This problem is exacerbated when considering multi-agent NN-controlled systems, as existing reachability methods often scale poorly for large systems. This paper addresses the problem of finding overapproximations of forward reachable sets for discretetime uncertain multi-agent systems with distributed NNC architectures. We first reformulate the dynamics, making the system more amenable to reachablility analysis. Next, we take advantage of the distributed architecture to split the overall reach ability problem into smaller problems, significantly reducing computation time. We use quadratic constraints, along with a convex representation of uncertainty in each agent's model, to form semidefinite programs, the solutions of which give overapproximations of forward reachable sets for each agent. Finally, the methodology is tested on two realistic examples: a platoon of vehicles and a power network system.