Researchers and engineers employ machine learning (ML) tools to detect pipe bursts and prevent significant non-revenue water losses in water distribution systems (WDS). Nonetheless, many approaches developed so far consider a fixed number of sensors, which requires the ML model redevelopment and collection of sufficient data with the new sensor configuration for training. To overcome these issues, this study presents a novel approach based on Long Short-Term Memory neural networks (NNs) that leverages transfer learning to manage a varying number of sensors and retain good detection performance with limited training data. The proposed detection model first learns to reproduce the normal behavior of the system on a dataset obtained in burst-free conditions. The training process involves predicting flow and pressure one-time step ahead using historical data and time-related features as inputs. During testing, a post-prediction step flags potential bursts based on the comparison between the observations and model predictions using a time-varied error threshold. When adding new sensors, we implement transfer learning by replicating the weights of existing channels and then fine-tune the augmented NN. We evaluate the robustness of the methodology on simulated fire hydrant bursts and real-bursts in 10 district metered areas (DMAs) of the UK. For real bursts, we perform a sensitivity analysis to understand the impact of data resolution and error threshold on burst detection performance. The results obtained demonstrate that this ML-based methodology can achieve Precision of up to 98.1% in real-life settings and can identify bursts, even in data scarce conditions.