Federated weather forecasting is a promising collaborative learning framework for analyzing meteorological data across participants from different countries and regions, thus embodying a global-scale real-time weather data predictive analytics platform to tackle climate change. This paper is to model the meteorological data in a federated setting where many distributed low-resourced sensors are deployed in different locations. Specifically, we model the spatial-temporal weather data into a federated prompt learning framework that leverages lightweight prompts to share meaningful representation and structural knowledge among participants. Prompts-based communication allows the server to establish the structural topology relationships among participants and further explore the complex spatial-temporal correlations without transmitting private data while mitigating communication overhead. Moreover, in addition to a globally shared large model at the server, our proposed method enables each participant to acquire a personalized model that is highly customized to tackle climate changes in a specific geographic area. We have demonstrated the effectiveness of our method on classical weather forecasting tasks by utilizing three spatial-temporal multivariate time-series weather data.