The collaboration between Unmanned Aerial Vehicles (UAVs) and Low Earth Orbit (LEO) satellites represents an emerging edge computing paradigm that holds significant potential for enabling ubiquitous connectivity in the Internet of Things (IoT). However, highly dynamic UAV-satellite links lead to high Doppler shift, high path loss, and other issues, which cause serious semantic distortion. Meanwhile, LEO satellite spectrum resources are extremely limited. Thus, realizing end-to-end reliable UAV-LEO satellite communications with lower communication bandwidth in poor channel environments presents a significant challenge. To address these challenges, this paper proposes the Semantics-driven Adaptive UAV-LEO Satellite Cooperative Communication System (SAULC), establishing the first UAV-LEO satellite semantic communication architecture. The system systematically incorporates time-varying channel characteristics, including Doppler shift, path loss, and atmospheric attenuation, while employing a low-complexity Flatten Transformer architecture with integrated channel feedback mechanisms. Experiments show that the proposed model significantly outperforms conventional schemes under time-varying UAV-satellite channels.