The assessment of chronic wound healing often depends on visual inspection, but subjective methods can introduce variability and hinder reliable predictions. Current computer-aided detection (CAD) tools encounter limitations in automated assessment, particularly in the fine-grained analysis of tissues with irregular textures. This paper introduces a new approach, a semi-supervised hierarchical convolutional neural network (SHCNN), for robust and efficient chronic wound assessment. SHCNN simultaneously handles two crucial tasks: accurate wound size measurement and fine-grained segmentation of seven tissue types. The model is initially trained using limited pixel-level annotations, and later relies on image-level labels, alleviating the need for scarce and arduously obtained pixel-level annotations. Evaluation on a dataset with 1200 image-level and 600 pixel-level annotations demonstrates the effectiveness of SHCNN. It achieves an average pixel-wise accuracy of 81% for tissue segmentation and measurement errors of 2% (small wounds) and 3% (large wounds). These findings suggest that SHCNN has the potential to replace manual assessment, providing a valuable tool for automated chronic wound management.