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Deep Neural Networks (DNNs) are becoming common in "learning-enabled" time-critical applications such as autonomous driving and robotics. One approach to protect DNN inference from adversarial actions and preserve model privacy/confidentiality is to execute them within trusted enclaves available in modern processors. However, running DNN inference inside limited-capacity enclaves while ensuring timing guarantees is challenging due to (a) large size of DNN workloads and (b) extra switching between "normal" and "trusted" execution modes. This paper introduces new time-aware scheduling schemes - DeepTrust^RT - to securely execute deep neural inferences for learning-enabled real-time systems. We first propose a variant of EDF (called DeepTrust^RT-LW) that slices each DNN layer and runs them sequentially in the enclave. However, due to extra context switch overheads of individual layer slices, we further introduce a novel layer fusion technique (named DeepTrust^RT-FUSION). Our proposed scheme provides hard real-time guarantees by fusing multiple layers of DNN workload from multiple tasks; thus allowing them to fit and run concurrently within the enclaves while maintaining real-time guarantees. We implemented and tested DeepTrust^RT ideas on the Raspberry Pi platform running OP-TEE+DarkNet-TZ DNN APIs and three DNN workloads (AlexNet-squeezed, Tiny Darknet, YOLOv3-tiny). Compared to the layer-wise partitioning approach (DeepTrust^RT-LW), DeepTrust^RT-FUSION can schedule up to 3x more tasksets and reduce context switches by up to 11.12x. We further demonstrate the efficacy of DeepTrust^RT using a flight controller (ArduPilot) case study and find that DeepTrust^RT-FUSION retains real-time guarantees where DeepTrust^RT-LW becomes unschedulable.