Collaborative perception (CP) is a critical technology in applications like autonomous driving and smart cities. It involves the sharing and fusion of information among sensors to overcome the limitations of individual perception, such as blind spots and range limitations. However, CP faces two primary challenges. First, due to the dynamic nature of the environment, the timeliness of the transmitted information is critical to perception performance. Second, with limited computational power at the sensors and constrained wireless bandwidth, the communication volume must be carefully designed to ensure feature representations are both effective and sufficient. This work studies the dynamic scheduling problem in a multi-region CP scenario, and presents a Timeliness-Aware Multi-region Prioritized (TAMP) scheduling algorithm to trade-off perception accuracy and communication resource usage. Timeliness reflects the utility of information that decays as time elapses, which is manifested by the perception performance in CP tasks. We propose an empirical penalty function that maps the joint impact of Age of Information (AoI) and communication volume to perception performance. Aiming to minimize this timeliness-oriented penalty in the long-term, and recognizing that scheduling decisions have a cumulative effect on subsequent system states, we propose the TAMP scheduling algorithm. TAMP is a Lyapunov-based optimization policy that decomposes the long-term average objective into a per-slot prioritization problem, balancing the scheduling worth against resource cost. We validate our algorithm in both intersection and corridor scenarios with the real-world Roadside Cooperative perception (RCooper) dataset. Extensive simulations demonstrate that TAMP outperforms the best-performing baseline, achieving an Average Precision (AP) improvement of up to 27% across various configurations.