Video compression is a standard procedure applied to all videos to minimize storage and transmission demands while preserving visual quality as much as possible. Therefore, evaluating the visual quality of compressed videos is crucial for guiding the practical usage and further development of video compression algorithms. Although numerous compressed video quality assessment (VQA) methods have been proposed, they often lack the generalization capability needed to handle the increasing diversity of video types, particularly high dynamic range (HDR) content. In this paper, we introduce CompressedVQA-HDR, an effective VQA framework designed to address the challenges of HDR video quality assessment. Specifically, we adopt the Swin Transformer and SigLip 2 as the backbone networks for the proposed full-reference (FR) and no-reference (NR) VQA models, respectively. For the FR model, we compute deep structural and textural similarities between reference and distorted frames using intermediate-layer features extracted from the Swin Transformer as its quality-aware feature representation. For the NR model, we extract the global mean of the final-layer feature maps from SigLip 2 as its quality-aware representation. To mitigate the issue of limited HDR training data, we pre-train the FR model on a large-scale standard dynamic range (SDR) VQA dataset and fine-tune it on the HDRSDR-VQA dataset. For the NR model, we employ an iterative mixed-dataset training strategy across multiple compressed VQA datasets, followed by fine-tuning on the HDRSDR-VQA dataset. Experimental results show that our models achieve state-of-the-art performance compared to existing FR and NR VQA models. Moreover, CompressedVQA-HDR-FR won first place in the FR track of the Generalizable HDR & SDR Video Quality Measurement Grand Challenge at IEEE ICME 2025. The code is available at https://github.com/sunwei925/CompressedVQA-HDR.