Rigorous and periodic monitoring of healing progression in chronic wounds is crucial for the proper management of this impactful health problem; however, manual assessment is highly subjective. Artificial Intelligence-based digital systems have arisen as a solution to automate the analysis of wound properties and reduce the variability of the assessment process at different levels. This study presents an automated approach for wound bed characterisation, using open wound detection and tissue segmentation algorithms to estimate the relative proportion of granulation, slough and eschar tissues. The impact of dataset composition on the performance of deep learning-based tissue segmentation models is investigated, along with the comparison of two model architectures (DeepLabV3+ with a ResNet50 backbone and UPerNet with a Swin Transformer backbone). Two private Wounds datasets are used and augmented with an external third-party external dataset. The results demonstrate that DeepLabV3+ outperforms UPerNet-Swin across all dataset combinations, establishing it as the preferred architecture in this case. Furthermore, incorporating all available datasets improves segmentation performance, underscoring the importance of data diversity. The best-performing model achieved Dice scores of 74.65%, 62.07% and 69% for granulation, slough, and eschar tissues, respectively, with corresponding mean absolute errors (MAE) of 16.51%, 14.68%, and 4.26% for tissue proportion estimation. A comparison of the standard deviation of the obtained MAE results with the ones reported for clinical experts in a similar task demonstrated that the proposed pipeline effectively decreased the variability of the estimated tissue percentages, providing a framework with the potential to streamline the wound monitoring process and increase its reproducibility.