We present a novel deep learning approach to categorical segmentation of lung CTs of COVID-19 patients. Specifically, we partition the scans into healthy lung tissues, non-lung regions, and two different, yet visually similar, pathological lung tissues, namely, ground-glass opacity and consolidation. This is accomplished via a unique, end-to-end hierarchical network architecture and ensemble learning, which contribute to the segmentation and provide a measure for segmentation uncertainty. The proposed framework achieves competitive results and outstanding generalization capabilities for three COVID-19 datasets. Our method is ranked second in a public Kaggle competition for COVID-19 CT images segmentation. Moreover, segmentation uncertainty regions are shown to correspond to the disagreements between the manual annotations of two different radiologists. Finally, preliminary promising correspondence results are shown for our private dataset when comparing the patients' COVID-19 severity scores (based on clinical measures), and the segmented lung pathologies. Code and data are available at our repository.