Many occupational cohort studies on underground miners have demonstrated that radon exposure is associated with an increased risk of lung cancer mortality. However, despite the deleterious consequences of exposure measurement error on statistical inference, these analyses traditionally do not account for exposure uncertainty. This might be due to the challenging nature of measurement error resulting from imperfect surrogate measures of radon exposure. Indeed, we are typically faced with exposure uncertainty in a time-varying exposure variable where both the type and the magnitude of error may depend on period of exposure. To address the challenge of accounting for multiplicative and heteroscedastic measurement error that may be of Berkson or classical nature, depending on the year of exposure, we opted for a Bayesian structural approach, which is arguably the most flexible method to account for uncertainty in exposure assessment. We assessed the association between occupational radon exposure and lung cancer mortality in the French cohort of uranium miners and found the impact of uncorrelated multiplicative measurement error to be of marginal importance. However, our findings indicate that the retrospective nature of exposure assessment that occurred in the earliest years of mining of this cohort as well as many other cohorts of underground miners might lead to an attenuation of the exposure-risk relationship. More research is needed to address further uncertainties in the calculation of lung dose, since this step will likely introduce important sources of shared uncertainty.