Monitoring and managing Earth's forests in an informed manner is an important requirement for addressing challenges like biodiversity loss and climate change. While traditional in situ or aerial campaigns for forest assessments provide accurate data for analysis at regional level, scaling them to entire countries and beyond with high temporal resolution is hardly possible. In this work, we propose a Bayesian deep learning approach to densely estimate forest structure variables at country-scale with 10-meter resolution, using freely available satellite imagery as input. Our method jointly transforms Sentinel-2 optical images and Sentinel-1 synthetic aperture radar images into maps of five different forest structure variables: 95th height percentile, mean height, density, Gini coefficient, and fractional cover. We train and test our model on reference data from 41 airborne laser scanning missions across Norway and demonstrate that it is able to generalize to unseen test regions, achieving normalized mean absolute errors between 11% and 15%, depending on the variable. Our work is also the first to propose a Bayesian deep learning approach so as to predict forest structure variables with well-calibrated uncertainty estimates. These increase the trustworthiness of the model and its suitability for downstream tasks that require reliable confidence estimates, such as informed decision making. We present an extensive set of experiments to validate the accuracy of the predicted maps as well as the quality of the predicted uncertainties. To demonstrate scalability, we provide Norway-wide maps for the five forest structure variables.