Quantum networks facilitate numerous applications including secure communication and distributed quantum computation by performing entanglement distribution. For some multiuser quantum applications, access to a shared multipartite state is required. We consider the problem of designing protocols for distributing such states, at an increased rate. For this, we propose three protocols that leverage multipath routing to increase the distribution rate for multiuser applications. The protocols are evaluated on quantum networks with noisy intermediate scale quantum (NISQ) constraints, including limited quantum memories and probabilistic entanglement generation. Simulation results show that the developed protocols achieve an exponential increase in the distribution rate of multipartite states compared to single-path routing techniques, with a maximum increase of four orders of magnitude for the cases studied. Furthermore, the relative increase in the distribution rate was also found to improve for larger sets of users. When the protocols were tested in scaled-down real-world topologies, it was found that a topology had a significant effect on the multipartite state distribution rates achieved by the protocols. Finally, we found that the benefits of multipath routing are maximum for short quantum memory decoherence times and intermediate values of entanglement generation probability. Hence, the protocols developed can benefit NISQ quantum network control and design.