Paediatric obstructive sleep apnoea (OSA) is clinically significant yet difficult to diagnose, as children poorly tolerate sensor-based polysomnography. Acoustic monitoring provides a non-invasive alternative for home-based OSA screening, but limited paediatric data hinders the development of robust deep learning approaches. This paper proposes a transfer learning framework that adapts acoustic models pretrained on adult sleep data to paediatric OSA detection, incorporating SpO2-based desaturation patterns to enhance model training. Using a large adult sleep dataset (157 nights) and a smaller paediatric dataset (15 nights), we systematically evaluate (i) single- versus multi-task learning, (ii) encoder freezing versus full fine-tuning, and (iii) the impact of delaying SpO2 labels to better align them with the acoustics and capture physiologically meaningful features. Results show that fine-tuning with SpO2 integration consistently improves paediatric OSA detection compared with baseline models without adaptation. These findings demonstrate the feasibility of transfer learning for home-based OSA screening in children and offer its potential clinical value for early diagnosis.