In disaster scenarios, ensuring both reliable communication and situational awareness becomes a critical challenge due to the partial or complete collapse of terrestrial networks. This paper proposes an integrated sensing and communication (ISAC) over non-terrestrial networks (NTN) architecture referred to as ISAC-over-NTN that integrates multiple uncrewed aerial vehicles (UAVs) and a high-altitude platform station (HAPS) to maintain resilient and reliable network operations in post-disaster conditions. We aim to achieve two main objectives: i) provide a reliable communication infrastructure, thereby ensuring the continuity of search-and-rescue activities and connecting people to their loved ones, and ii) detect users, such as those trapped under rubble or those who are mobile, using a Doppler-based mobility detection model. We employ an innovative beamforming method that simultaneously transmits data and detects Doppler-based mobility by integrating multi-user multiple-input multiple-output (MU-MIMO) communication and monostatic sensing within the same transmission chain. The results show that the proposed framework maintains reliable connectivity and achieves high detection accuracy of users in critical locations, reaching 90% motion detection sensitivity and 88% detection accuracy.