This paper proposes a Distributed Moving Horizon Estimation (DMHE) with an Extended Kalman Filter (EKF)- based pre-estimation to solve the constrained cooperative localization problem for a Multi-Agent System (MAS) using nonlinear measurements. The proposed DMHE strategy uses a fused arrival cost obtained by a consensus among neighbors to efficiently spread the relevant estimation information across the communication network. The EKF pre-estimation enables to reduce the number of optimization variables and, thus, the computation time of the constrained nonlinear optimization problem over the horizon length, while preserving the accuracy of the estimation. A simulation case study of cooperative localization of a fleet of Unmanned Aerial Vehicles (UAVs) is proposed. Comparison with existing distributed estimation methods is carried out to confirm the effectiveness of the proposed DMHE algorithm in terms of estimation accuracy, computation time, and constraints handling.