This paper proposes a nonparametric approach to detecting changes in variance within a time series that we demonstrate is resilient to departures from the assumption of normality or presence of outliers. Our method is founded on a local estimate of the variance provided by the locally stationary wavelet framework. Within this setting, the structure of this local estimate of the variance will be piecewise constant if a time series has piecewise constant variance. Consequently, changes in the variance of a time series can be detected in a nonparametric setting. In addition, using a simulation study, we explore the robustness of our approach against the typical assumption of normality and presence of outliers. We illustrate the application of the approach to changes in variability of wind speeds at a location in the United Kingdom.