Accurate probabilistic prediction of wind power is crucial for maintaining grid stability and facilitating the efficient integration of renewable energy sources. Gaussian process (GP) models offer a principled framework for quantifying uncertainty; however, conventional approaches typically rely on stationary kernels and homoscedastic noise assumptions, which are inadequate for modelling the inherently non-stationary and heteroscedastic nature of wind speed and power output. We propose a heteroscedastic non-stationary GP framework based on the generalised spectral mixture kernel, enabling the model to capture input-dependent correlations as well as input-dependent variability in wind speed-power data. We evaluate the proposed model on 10-minute supervisory control and data acquisition (SCADA) measurements and compare it against GP variants with stationary and non-stationary kernels, as well as commonly used non-GP probabilistic baselines. The results highlight the necessity of modelling both non-stationarity and heteroscedasticity in wind power prediction and demonstrate the practical value of flexible non-stationary GP models in operational SCADA settings.