Grant-free non-coherent index-modulation (NC-IM) has been recently considered as an efficient massive access scheme for enabling cost-and energy-limited Internet-of-Things (IoT) devices that transmit small data packets. This paper investigates the grant-free NC-IM scheme combined with orthogonal frequency division multiplexing for applicant to unmanned aerial vehicle (UAV)-based massive IoT access. Specifically, each device is assigned a unique non-orthogonal signature sequence codebook. Each active device transmits one of its signature sequences in the given time-frequency resources, by modulating the information in the index of the transmitted signature sequence. For small-scale multiple-input multiple-output (MIMO) deployed at the UAV-based aerial base station (BS), by jointly exploiting the space-time-frequency domain device activity, we propose a computationally efficient space-time-frequency joint activity and blind information detection (JABID) algorithm with significantly improved detection performance. Furthermore, for large-scale MIMO deployed at the aerial BS, by leveraging the sparsity of the virtual angular-domain channels, we propose an angular-domain based JABID algorithm for improving the system performance with reduced access latency. In addition, for the case of high mobility IoT devices and/or UAVs, we introduce a timefrequency spread transmission (TFST) strategy for the proposed JABID algorithms to combat doubly-selective fading channels. Finally, extensive simulation results are illustrated to verify the superiority of the proposed algorithms and the TFST strategy over known state-of-the-art algorithms.