A multistage manufacturing system with multiple manufacturing stages is the key and main production mode for enterprises to achieve lean production. Due to the variation propagation between stages and multiple related quality prediction tasks, it is difficult to accurately predict the quality of multistage manufacturing systems with multiple tasks. Traditional single stage and single task quality prediction methods permit the multi stages and multiple tasks separately, which ignore multi-stage effects or the quality-related relationship between multiple quality output indicators and reduce the efficiency of quality prediction. In this paper, a production quality prediction framework based on multi-task joint deep learning is proposed to simultaneously evaluate the multi-task quality of all stages in a multistage manufacturing system. To be specific, variation propagation cumulative impact between multiple manufacturing stages is innovatively expressed by designing a multi-scale convolutional neural network with control gates (MCNN-CG) to extract and propagate data features. Production quality with multi-tasks at all stages is jointly predicted by designing a multi-layer multi-gate mixture-of-experts multi-task (ML-MMoE) model with reducing multi-task predictive loss simultaneously. The soft parameter-sharing strategy and multi-gate attention strategy are separately designed to ensure information sharing while learning personalized features of tasks to improve quality prediction accuracy of each task. In addition, a loss function based on homoscedastic uncertainty and regularization is designed to automatically learn the weight between multi-stage and multi-task losses. Experiments on multistage assembly test data of an inertial navigation manufacturing system show that the proposed method performs better than traditional models. Compared to the single-stage model, the proposed multistage model has an average improvement of 8.5%, 20.0% and 23.3% in R2, MAE and RMSE respectively in the second stage. Compared with the traditional multi-stage model, the proposed model has an average improvement of 1.7%, 6.2% and 9.8% in R2, MAE and RMSE respectively in the second stage.