Uncertainty quantification in digital twins is critical to enable reliable and credible predictions beyond available data. A key challenge is that ensemble-based approaches can become prohibitively expensive when embedded in control and data assimilation loops in digital twins, even when reduced models are used. We introduce a reduced modeling approach that advances in time the mean and covariance of the reduced solution distribution induced by the initial condition uncertainties, which eliminates the need to maintain and propagate a costly ensemble of reduced solutions. The mean and covariance dynamics are obtained as a moment closure from Neural Galerkin schemes on pre-trained neural networks, which can be interpreted as filtered Neural Galerkin dynamics analogous to Gaussian filtering and the extended Kalman filter. Numerical experiments demonstrate that filtered Neural Galerkin schemes achieve more than one order of magnitude speedup compared to ensemble-based uncertainty propagation.