Real-time state estimation underpins advanced applications in distribution networks by revealing system operating conditions with high fidelity. Intelligent measurement terminals increasingly combine heterogeneous measurements, including micro-phasor measurement units (μPMUs), remote terminal units (RTUs), and data transmission units (DTUs). Their complementary strengths can overcome traditional limits in time resolution and accuracy. However, heterogeneous formats, asynchronous updates, and complex temporal dynamics challenge conventional estimators. This paper proposes a unified real-time forecasting-aided state estimation (FASE) framework for distribution networks with multi-source measurement data fusion that fuses μPMU, RTU, and DTU data. A long short-term memory (LSTM)-based data imputation method is designed to effectively align delayed RTU and DTU updates with μPMU sampling, markedly reducing imputation errors versus linear and historical-average baselines. Unified measurement equations are formulated for all devices. An improved cubature Kalman filter (CKF) with adaptive robust weighting is adopted to enhance numerical stability and outlier resilience. To capture multimodal operating regimes driven by variable distributed energy resources, a Gaussian mixture model (GMM)-based approach is integrated into the distribution network state transition. Validated on the IEEE 33-bus and 118-bus systems, the proposed FASE framework achieves higher estimation accuracy and computational efficiency than extended Kalman filter (EKF), unscented Kalman filter (UKF) and traditional CKF while meeting real-time constraints.