In this work, we study stability of distributed filtering of Markov chains with finite state space, partially observed in conditionally Gaussian noise. We consider a nonlinear filtering scheme over a Distributed Network of Agents (DNA), 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 for each sensor in the network, we fully characterize a minimal number of such steps, such that the distributed filter remains uniformly stable with a prescribed accuracy level, {\varepsilon} \in (0,1], within a finite operational horizon, T, and across all sensors. Stability is in the sense of the \ell_1-norm between the centralized and distributed versions of the posterior at each sensor, and at each time within T. Roughly speaking, our main result shows that uniform {\varepsilon}-stability of the distributed filtering process depends only loglinearly on T and (roughly) the size of the network, and only logarithmically on 1/{\varepsilon}. If this total loglinear bound is fulfilled, any additional consensus iterations will incur a fully quantified further exponential decay in the consensus error. Our bounds are universal, in the sense that they are independent of the particular structure of the Gaussian Hidden Markov Model (HMM) under consideration.