In real-world time-series modelling, graph structures are widely adopted because they explicitly encode node topology and capture complex network dynamics. In practice, however, a complete graph is often partitioned across multiple parties; each party can access only its local sub-graph and, owing to privacy regulations, cannot share topology or data, creating pervasive data silos. Federated Graph Learning (FGL) offers a privacy-preserving collaborative-learning paradigm, yet current methods still face two key challenges: (1) they implicitly capture inter-edge information, making it difficult to accurately reconstruct the global structure and consequently degrading model performance; (2) explicitly exchanging inter-edge information may leak graph-topology privacy. To overcome these obstacles, we propose FedSkeleton, a privacy-preserving framework for time-series prediction that comprises a Skeleton Construction Module and a Dual-stream Forecasting Module, enabling global dependency capture without revealing the topology. Extensive experiments show that FedSkeleton consistently outperforms existing baselines and even surpasses centralised models with full-graph access. In addition, we conduct comprehensive security analysis, communication-cost evaluation and scalability experiments, demonstrating that FedSkeleton effectively resists common attacks, keeps communication overhead manageable and remains robust with respect to key hyper-parameters and the number of participating parties.