Time series data widely exist in public services, industrial environments, and military applications. Traditionally, the transmission of a huge volume of data for analytic tasks poses challenges, particularly in mobile environments with limited computing and communication resources. Semantic communication emerges as a solution for intelligently extracting various features from source data and efficiently transmitting task-related information to receivers, thereby reducing bandwidth consumption significantly. In this paper, we introduce a novel federated semantic communication system tailored for forecasting-oriented time series transmission tasks. The correlation of source data collected from terminal devices is mined and the corresponding semantic information is transmitted to an edge server for collaborative inference. To optimize the semantic analysis process, we devise a deep decomposition block at the transmitter side, decomposing time series into trend and multiple period components. This reduces noise interference from wireless channels, enhancing the overall transmission quality. For effective training and collaborative inference, we propose a Federated Mixture of period Routers (FedMoR) architecture. Within each channel encoder, period routers are divided into private and public ones. Private routers extract specialized features from individually collected data, mitigating accuracy degradation. Public routers share knowledge across all transmitters, enhancing temporal analysis robustness. Simulation results demonstrate that the proposed system outperforms two traditional technique-based and two semantic communication-based baselines under three common channels. The system achieves low mean square errors on five widely-used real-world time series forecasting datasets, particularly in the low signal-to-noise ratio regime.