This paper addresses the problem of cooperative object transportation for multiple Underwater Vehicle Manipulator Systems (UVMSs) in a constrained workspace involving static obstacles. We propose a Nonlinear Model Predictive Control (NMPC) approach for a team of UVMSs in order to transport an object while avoiding significant constraints and limitations such as: kinematic and representation singularities, obstacles within the workspace, joint limits and control input saturations. More precisely, by exploiting the coupled dynamics between the robots and the object, and using certain load sharing coefficients, we design a distributed NMPC for each UVMS in order to cooperatively transport the object within the workspace's feasible region. Moreover, the control scheme adopts load sharing among the UVMSs according to their specific payload capabilities. Additionally, the feedback relies on each UVMS's locally measurements and no explicit data is exchanged online among the robots, thus reducing the required communication bandwidth. Finally, real-time simulation results conducted in UwSim dynamic simulator running in ROS environment verify the efficiency of the theoretical finding.