Artificial intelligence (AI) has demonstrated considerable potential in the realm of medical imaging. However, the development of high-performance AI models typically necessitates training on large-scale, centralized datasets. This approach is confronted with significant challenges due to strict patient privacy regulations and legal restrictions on data sharing and utilization. These limitations hinder the development of large-scale models in medical domains and impede continuous updates and training with new data. Federated Learning (FL), a privacy-preserving distributed training framework, offers a new solution by enabling collaborative model development across fragmented medical datasets. In this survey, we review FL's contributions at two stages of the full-stack medical analysis pipeline. First, in upstream tasks such as CT or MRI reconstruction, FL enables joint training of robust reconstruction networks on diverse, multi-institutional datasets, alleviating data scarcity while preserving confidentiality. Second, in downstream clinical tasks like tumor diagnosis and segmentation, FL supports continuous model updating by allowing local fine-tuning on new data without centralizing sensitive images. We comprehensively analyze FL implementations across the medical imaging pipeline, from physics-informed reconstruction networks to diagnostic AI systems, highlighting innovations that improve communication efficiency, align heterogeneous data, and ensure secure parameter aggregation. Meanwhile, this paper provides an outlook on future research directions, aiming to serve as a valuable reference for the field's development.