Power production of offshore wind farms depends on many parameters and is significantly affected by wake losses. Due to the intermittency of wind power and its rapidly increasing share in the total energy mix, accurate forecasting of wind farm power production becomes increasingly important. This paper presents a data-driven methodology for forecasting power production and wake losses of wind farms, taking the dynamics of weather conditions into account. A modular approach is adopted by integrating multiple deep neural networks, resulting in a digital twin of the wind farm that can be interfaced with weather forecasts of different meteorological service providers. Another key advantage of the employed data-driven approach is its high prediction speed compared to physics-based methods, such that it can be employed for applications where real-time power forecasting is required. The methodology has been applied to two large offshore wind farms located within the Belgian-Dutch wind farm cluster in the North Sea.