Goal-oriented communication has become one of the focal concepts in sixth-generation communication systems owing to its potential to provide intelligent, immersive, and real-time mobile services. The emerging paradigms of goal-oriented communication constitute closed loops integrating communication, computation, and sensing. However, challenges arise for closed-loop timing analysis due to multiple random factors that affect the communication/computation latency, as well as the heterogeneity of feedback mechanisms across multi-modal sensing data. To tackle these problems, we aim to provide a unified timing analysis framework for closed-loop goal-oriented communication (CGC) systems over fading channels. The proposed framework is unified as it considers computation, compression, and communication latency in the loop with different configurations. To capture the heterogeneity across multi-modal feedback, we categorize the sensory data into the periodic-feedback and event-triggered, respectively. We formulate timing constraints based on average and tail performance, covering timeliness, jitter, and reliability of CGC systems. A method based on saddlepoint approximation is proposed to obtain the distribution of closed-loop latency. The results show that the modified saddlepoint approximation is capable of accurately characterizing the latency distribution of the loop with analytically tractable expressions. This sets the basis for low-complexity co-design of communication and computation.