Monitoring the Covid19 pandemic is critical to design sanitary policies. Recently, reliable estimates of the pandemic reproduction number were obtained from a nonsmooth convex optimization procedure designed to fit epidemiology requirements and to be robust to the low quality of the data (outliers, pseudo-seasonalities, \ldots). Applied to daily new infection counts made public by National Health Agencies and centralized by Johns Hopkins University, robust estimates of the reproduction number for 200+ countries are updated and published every day. To further improve estimation procedures and also, and mostly, increase their usability by epidemiologists, the present work exploits the Bayesian paradigm and derives a new Monte Carlo method to sample from a nonsmooth convex a posteriori distribution. This new sampler stems from an original combination of the Langevin Monte Carlo algorithm with Proximal operators. Its relevance and practical efficiency to produce meaningful credibility intervals for the Covid19 reproduction number are assessed from several indices quantifying the statistics of the Monte Carlo chains, and making use of real daily new infection counts.