In this paper we study the problem of characterizing and computing the nonanticipative rate distortion function (NRDF) for partially observable multivariate Gauss-Markov processes with hard mean squared error (MSE) distortion constraints. For the finite time horizon case, we first derive the complete characterization of this problem and its corresponding optimal realization which is shown to be a linear functional of the current time sufficient statistic of the past and current observations signals. We show that when the problem is strictly feasible, it can be computed via semidefinite programming (SDP) algorithm. For time-varying scalar processes with average total MSE distortion we derive an optimal closed form expression by means of a dynamic reverse-waterfilling solution that we also implement via an iterative scheme that convergences linearly in finite time, and a closed-form solution under pointwise MSE distortion constraint. For the infinite time horizon, we give necessary and sufficient conditions to sure that asymptotically the sufficient statistic process of the observation signals achieves a steady-state solution for the corresponding covariance matrices and impose conditions that allow existence of a time-invariant solution. Then, we show that when a finite solution exists in the asymptotic limit, it can be computed via SDP algorithm. We also give strong structural properties on the characterization of the problem in the asymptotic limit that allow for an optimal solution via a reverse-waterfilling algorithm that we implement via an iterative scheme that converges linearly under a finite number of spatial components. Subsequently, we compare the computational time needed to execute for both SDP and reverse-waterfilling algorithms when these solve the same problem to show that the latter is a scalable optimization technique. Our results are corroborated with various simulation studies and are also compared with existing results in the literature.