Integrating deep learning with environmental perception enhances robotic adaptability to complex tasks. However, its “black-box” nature, such as the lack of uncertainty quantification, poses challenges for safety-critical applications, particularly in unstructured and noisy environments. Bayesian neural networks (BNNs) offer uncertainty quantification but are limited by high hardware overhead, restricting real-time implementation on resource-constrained robots. This paper presents a mixedsignal hardware accelerator for BNNs, utilizing probabilistic quantum tunneling in fully depleted silicon-on-insulator (FDSOI) transistors to enable efficient, real-time uncertainty quantification. Device measurements indicate high-quality Gaussian random variable generation, validated through quantile-quantile plot analysis, with a high correlation coefficient ($r=0.997$) at $200 \mathrm{fJ} /$ sample. Leveraging such compact randomness, the parallel architecture achieved $10^{3}-10^{4} \times$ latency reduction at less than $2 \times$ area cost. Finally, in uncertainty-aware visual localization application of autonomous underwater vehicles, the BNN model effectively distinguishes data noise from model uncertainty, yielding significant information gain and enhancing the resampling efficiency by $4.5 \times$ at same accuracy.