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With the rapid development of cloud manufacturing, industrial production with edge computing as the core architecture has been greatly developed. However, edge devices often suffer from abnormalities and failures in industrial production. Therefore, detecting these abnormal situations timely and accurately is crucial for cloud manufacturing. As such, a straightforward solution is that the edge device uploads the data to the cloud for anomaly detection. However, Industry 4.0 puts forward higher requirements for data privacy and security so that it is unrealistic to upload data from edge devices directly to the cloud. Considering the above-mentioned severe challenges, this paper customizes a weakly-supervised edge computing anomaly detection framework, i.e., Federated Learning-based Transformer framework (\textit{FedAnomaly}), to deal with the anomaly detection problem in cloud manufacturing. Specifically, we introduce federated learning (FL) framework that allows edge devices to train an anomaly detection model in collaboration with the cloud without compromising privacy. To boost the privacy performance of the framework, we add differential privacy noise to the uploaded features. To further improve the ability of edge devices to extract abnormal features, we use the Transformer to extract the feature representation of abnormal data. In this context, we design a novel collaborative learning protocol to promote efficient collaboration between FL and Transformer. Furthermore, extensive case studies on four benchmark data sets verify the effectiveness of the proposed framework. To the best of our knowledge, this is the first time integrating FL and Transformer to deal with anomaly detection problems in cloud manufacturing.