Most time series observed in practice exhibit first as well as second-order nonstationarity. In this article we propose a novel framework for modelling series with simultaneous time-varying first and second-order structure, removing the restrictive zero-mean assumption of locally stationary wavelet processes and extending the applicability of the locally stationary wavelet model to include trend components. We develop an associated estimation theory for both first and second order time series quantities and show that our estimators achieve good properties in isolation of each other by making appropriate assumptions on the series trend. We demonstrate the utility of the method by analysing the global mean sea temperature time series, highlighting the impact of the changing climate.