Along with rich health-related metadata, medical images have been acquired for over 40,000 male and female UK Biobank participants, aged 44-82, since 2014. Phenotypes derived from these images, such as measurements of body composition from MRI, can reveal new links between genetics, cardiovascular disease, and metabolic conditions. In this work, six measurements of body composition and adipose tissues were automatically estimated by image-based, deep regression with ResNet50 neural networks from neck-to-knee body MRI. Despite the potential for high speed and accuracy, these networks produce no output segmentations that could indicate the reliability of individual measurements. The presented experiments therefore examine uncertainty quantification with mean-variance regression and ensembling to estimate individual measurement errors and thereby identify potential outliers, anomalies, and other failure cases automatically. In 10-fold cross-validation on data of about 8,500 subjects, mean-variance regression and ensembling showed complementary benefits, reducing the mean absolute error across all predictions by 12%. Both improved the calibration of uncertainties and their ability to identify high prediction errors. With intra-class correlation coefficients (ICC) above 0.97, all targets except the liver fat content yielded relative measurement errors below 5%. Testing on another 1,000 subjects showed consistent performance, and the method was finally deployed for inference to 30,000 subjects with missing reference values. The results indicate that deep regression ensembles could ultimately provide automated, uncertainty-aware measurements of body composition for more than 120,000 UK Biobank neck-to-knee body MRI that are to be acquired within the coming years.