Starting from Navier-Stokes' equation we derive two shallow water multilayer models for yield stress fluids, depending on the asymptotic analysis. One of them takes into account the normal stress contributions, making possible to recover a pseudoplug layer instead of a purely plug zone. A specific numerical scheme is designed to solve this model thanks to a finite volume discretization. It involves well-balancing techniques to be able to compute accurately the transitions between yielded and unyielded (or pseudoplug) zones, an important feature of the original partial differential equations' model. We perform numerical simulations on various test cases relevant to this physics: analytical solution of a uniform flow, steady solutions for arrested state, and a viscoplastic dam break. Simulations agree well when we perform comparisons with physical experiments of the group of Christophe Ancey (EPFL) and we make a comparative study including shallow water models and lubrication models that they present in [Ancey et al. Viscoplastic dam break waves: Review of simple computational approaches and comparison with experiments. Advances in Water Resources 48 (2012) 79-91]. Thanks to the multilayer structure of our model, we can go further on the description of the vertical structure associated to the (bottom) sheared layer and the top (pseudo-)plug layer.