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Residential electricity consumption forecasting plays a crucial role in the rational allocation of resources reducing energy waste and enhancing the grid-connected operation of power systems. Probabilistic forecasting can provide more comprehensive information for the decision-making and dispatching process by quantifying the uncertainty of electricity load. In this study, we propose a method based on stationary wavelet transform (SWT), quantile regression (QR), Bidirectional nested long short-term memory (BiNLSTM), and Depthwise separable convolution (DSC) combined with attention mechanism for electricity consumption probability prediction methods. First, the data sequence is decomposed using SWT to reduce the complexity of the sequence; then, the combined neural network model with attention is used to obtain the prediction values under different quantile conditions. Finally, the probability density curve of electricity consumption is obtained by combining kernel density estimation (KDE). The model was tested using historical demand-side data from five UK households to achieve energy consumption predictions 5 min in advance. It is demonstrated that the model can achieve both reliable probabilistic prediction and accurate deterministic prediction.