Existing work has shown that analysing differentially private (DP) synthetic data as if it were real does not produce valid uncertainty estimates. We tackle this problem by combining synthetic data analysis techniques from the field of multiple imputation (MI), and synthetic data generation using a novel noise-aware (NA) synthetic data generation algorithm NAPSU-MQ into a pipeline NA+MI that allows computing accurate uncertainty estimates for population-level quantities from DP synthetic data. Our experiments demonstrate that the pipeline is able to produce accurate confidence intervals from DP synthetic data.