Reconstructing bandlimited graph signals from a subset of noisy measurements is a fundamental challenge within the realm of signal processing. Historically, this problem has been approached assuming uniform noise variance across the network. Nevertheless, practical scenarios often present heterogeneous noise landscapes, greatly complicating the signal reconstruction process. This study tackles reconstruction of graph signals across networks where measurements may be affected by heterogeneous noise. A Bayesian model tailored for graph signals is employed, considering the potential existence of node-specific variations in measurement variance, namely different (and unknown) levels of uncertainty. Moreover, a novel uncertainty-aware local graph coherence metric is introduced, capitalizing on estimated parameters to refine the sampling process. By accommodating uncertainty, signal reconstruction accuracy is enhanced, even in demanding noise conditions. The proposed approach revolves around a framework combining maximum likelihood and maximum a-posteriori principles. Specifically, each observation is weighted based on a soft classification of nodes, so incorporating measurements reliability into the reconstruction process. The latter is performed through a novel algorithm coupling re-weighted iterative least squares with expectation-maximization. Such an algorithm can effectively manage heterogeneous noise and features a non-local regularization term, which promotes sparsity in the reconstructed signal while preserving signal discontinuities, crucial for capturing the characteristics of the underlying graph signal. Extensive simulations demonstrate the effectiveness of the proposed approach for various graph topologies and anomalous conditions, revealing substantial enhancements in signal reconstruction compared to existing methods. An illustrative example on PM10 data from the European Copernicus Atmosphere Monitoring Service (CAMS) is also reported.