Autonomous navigation of drones using computer vision has achieved promising performance. Nano-sized drones based on edge computing platforms are lightweight, flexible, and cheap; thus, they are suitable for exploring narrow spaces. However, due to their extremely limited computing power and storage, vision algorithms designed for high-performance GPU platforms cannot be used for nano-drones. To address this issue, this paper presents a lightweight CNN depth estimation network deployed on nano-drones for obstacle avoidance. Inspired by knowledge distillation (KD), a Channel-Aware Distillation Transformer (CADiT) is proposed to facilitate the small network to learn knowledge from a larger network. The proposed method is validated on the KITTI dataset and tested on a Crazyflie nano-drone with an ultra-low power microprocessor GAP8. This paper also implements a communication pipe so that the collected images can be streamed to a laptop through the on-board Wi-Fi module in real-time, enabling an offline reconstruction of the environment.