The global installed capacity of floating offshore wind turbines is projected to increase by at least 100 times over the next decades. Station-keeping of floating offshore renewable energy devices is achieved through the use of mooring systems. Mooring systems are exposed to a variety of environmental and operational conditions that cause corrosion, abrasion, and fatigue. Regular physical in-service inspections of mooring systems are the golden standard for monitoring their health status. This approach is often expensive, inefficient, and unsafe, and for this reason, researchers are focusing on developing tools for digital solutions for real-time monitoring. Floating offshore renewable energy devices are usually equipped with a wide range of sensors, some low-cost, low/zero maintenance, and easily deployable (e.g., accelerometers on the tower), contrary to others (e.g., direct tension mooring line measurements), producing real-time data streams. In this paper, we propose exploiting the data coming from the first type of sensors for mooring systems health status nowcasting. In particular, we will first rely on state-of-the-art supervised shallow and deep learning models for predicting the health status of the different sections of the mooring lines. Then, since these supervised models require types and amount of data that are seldom available, we will propose new shallow and deep weekly supervised models that require a very small amount of data regarding worn mooring lines. Results will show that these last models can potentially have practical applicability and impact for real-time monitoring of mooring systems in the near future. In order to support our statements, we will make use of data generated with a state-of-the-art digital twin of the mooring system, OrcaFlex, for a floating offshore wind turbine reproducing the physical mechanism of the mooring degradation under different loads and environmental conditions. Results will show errors around 1% in the simplest scenario and errors around 4% in the most challenging one, confirming the potentiality of the proposed approaches.