We derive the finite-time horizon nonanticipative rate distortion function (NRDF) of timevarying scalar Gauss-Markov sources under an average mean squared-error (MSE) distortion fidelity. Further, we show that a conditionally Gaussian reproduction process realizes the optimal reproduction distribution, and this is determined from the solution of a dynamic reversewaterfilling optimization problem. We provide an iterative algorithm that approximates the solution of the dynamic reverse-waterfilling problem. From the above results, we also obtain, as a special case, the NRDF under a per-letter or pointwise MSE distortion fidelity, and we draw connections to the classical RDF of Gaussian processes. Our results are corroborated with illustrative examples.