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Reproducibility has emerged as a critical bottleneck in data-driven materials engineering, particularly as graph neural networks (GNNs) and associated uncertainty quantification (UQ) methods are increasingly embedded in high-stakes discovery pipelines. While advances in conformal prediction, Bayesian inference, and ensemble techniques have improved predictive reliability, their practical deployment remains constrained by fragmented workflows, opaque data provenance, and inconsistent reporting standards. This review reframes uncertainty-aware materials modeling through the lens of reproducible workflows, tracing the pipeline from dataset construction and curation to model training, calibration, and deployment. Drawing on peer-reviewed studies, we integrate methodological advances in UQ with emerging practices in data governance, experiment tracking, and model documentation. The analysis reveals that uncertainty estimates are only as trustworthy as the workflows that generate them: biases in dataset composition, undocumented preprocessing steps, and inconsistent calibration protocols systematically undermine reliability, even when state-of-the-art UQ methods are applied. We argue that reproducibility must be treated as a first-class design constraint, requiring standardized data provenance tracking, version-controlled training pipelines, and model cards that explicitly document uncertainty behavior, calibration performance, and failure modes. By linking UQ theory with reproducible systems design, this work establishes a framework for trustworthy materials graph learning in which uncertainty is not merely computed but auditable, interpretable, and transferable across applications.