In this work, we study stability of distributed non-linear filtering of Markov chains with finite state space, partially observed in conditionally Gaussian noise. We propose a filtering scheme, which relies on the distributed evaluation of the likelihood part of the centralized nonlinear filter and is based on a particular specialization of the Alternating Direction Method of Multipliers (ADMM) for fast average consensus. Assuming the same number of consensus steps between any two consecutive noisy measurements, our main contribution is summarized in the full characterization of a minimal number of iterations, such that the distributed filter remains uniformly stable with a prescribed accuracy level, within a finite operational horizon, T and across all sensors. Our main result shows that e-stability of the distributed filtering process depends only loglinearly on T and (roughly) the size of the network. If this loglinear bound is fulfilled, any additional consensus iterations will further incur a fully quantified exponential decay in the consensus error. Our bounds are universal, in the sense that they are independent of the structure of the HMM under consideration.