In this work, we propose a fully Bayesian uncertainty analysis of the indirect measurement of thermal properties of walls from in situ temperature and flux measurements, obtained with an active method, using a one dimensional transient thermal model. We show that this approach is able to take into account the uncertainty of the inputs of the thermal model and the uncertainty of the output observations, for a more reliable uncertainty estimation of the calibration parameters and any derived quantity. For this problem, we improve the classical Bayesian inversion model by taking into account underestimated uncertainty on reported output observations, which is a frequently encountered issue in practice. We provide some recommendations for a wider applicability of the method. We illustrate the principles of uncertainty evaluation of the Guide to the Expression of Uncertainty in Measurement in terms of a real case study to evaluate the thermal resistance of a multilayer wall placed in a climatic chamber. For this application, we compare results of the Bayesian inversion with classical steady-state results in comparable experimental conditions. We perform a sensitivity analysis to study the effect of duration, input uncertainties and excess variance prior, and we make recommendations. R code is made available that enables a Bayesian uncertainty evaluation of inversion models for related applications.