Source-free domain adaptation (SFDA) alleviates the domain discrepancy among data obtained from domains without accessing the data for the awareness of data privacy. However, existing conventional SFDA methods face inherent limitations in medical contexts, where medical data are typically collected from multiple institutions using various equipment. To address this problem, we propose a simple yet effective method, named Uncertainty-aware Adaptive Distillation (UAD) for the multi-source-free unsupervised domain adaptation (MSFDA) setting. UAD aims to perform well-calibrated knowledge distillation from (i) model level to deliver coordinated and reliable base model initialisation and (ii) instance level via model adaptation guided by high-quality pseudo-labels, thereby obtaining a high-performance target domain model. To verify its general applicability, we evaluate UAD on two image-based diagnosis benchmarks among two multi-centre datasets, where our method shows a significant performance gain compared with existing works. The code will be available soon.