Many multivariate time series observed in practice are second order nonstationary, i.e. their covariance properties vary over time. In addition, missing observations in such data are encountered in many applications of interest, due to recording failures or sensor dropout, hindering successful analysis. This article introduces a novel method for data imputation in multivariate nonstationary time series, based on the so-called locally stationary wavelet modelling paradigm. Our methodology is shown to perform well across a range of simulation scenarios, with a variety of missingness structures, as well as being competitive in the stationary time series setting. We also demonstrate our technique on data arising in a health monitoring application.