Changes in model dynamics due to factors like actuator faults, platform aging, and unexpected disturbances can challenge an autonomous robot during real-world operations affecting its intended behavior and safety. Under such circumstances, it becomes critical to improve tracking performance, predict future states of the system, and replan to maintain safety and liveness conditions. In this letter, we propose a meta-learning-based framework to learn a model to predict the future system's states and their uncertainties under unforeseen and untrained conditions. Meta-learning is considered for this problem thanks to its ability to easily adapt to new tasks with a few data points gathered at runtime. We use the predictions from the meta-learned model to detect unsafe situations and proactively replan the system's trajectory when an unsafe situation is detected (e.g., a collision with an object). The proposed framework is applied and validated with both simulations and experiments on a faulty UAV performing an infrastructure inspection mission, demonstrating safety improvements.