In the realm of autonomous vehicle (AV) perception, comprehending 3D scenes is paramount for tasks such as planning and mapping. Semantic scene completion (SSC) aims to infer scene geometry and semantics from limited observations. While camera-based SSC has gained popularity due to affordability and rich visual cues, existing methods often neglect the inherent uncertainty in models. To address this, we propose an uncertainty-aware camera-based 3D semantic scene completion method ($\alpha$-SSC). Our approach includes an uncertainty propagation framework from depth models (Depth-UP) to enhance geometry completion (up to 11.58% improvement) and semantic segmentation (up to 14.61% improvement). Additionally, we propose a hierarchical conformal prediction (HCP) method to quantify SSC uncertainty, effectively addressing high-level class imbalance in SSC datasets. On the geometry level, we present a novel KL divergence-based score function that significantly improves the occupied recall of safety-critical classes (45% improvement) with minimal performance overhead (3.4% reduction). For uncertainty quantification, we demonstrate the ability to achieve smaller prediction set sizes while maintaining a defined coverage guarantee. Compared with baselines, it achieves up to 85% reduction in set sizes. Our contributions collectively signify significant advancements in SSC accuracy and robustness, marking a noteworthy step forward in autonomous perception systems.