Turbine wake and local blockage effects are known to alter wind farm power production in two different ways: (1) by changing the wind speed locally in front of each turbine; and (2) by changing the overall flow resistance in the farm and thus the so-called farm blockage effect. To better predict these effects with low computational costs, we develop data-driven emulators of the ‘local’ or ‘internal’ turbine thrust coefficient C_T^* as a function of turbine layout. We train the model using a multi-fidelity Gaussian Process (GP) regression with a combination of low (engineering wake model) and high-fidelity (Large-Eddy Simulations) simulations of farms with different layouts and wind directions. A large set of low-fidelity data speeds up the learning process and the high-fidelity data ensures a high accuracy. The trained multi-fidelity GP model is shown to give more accurate predictions of C_T^* compared to a standard (single-fidelity) GP regression applied only to a limited set of high-fidelity data. We also use the multi-fidelity GP model of C_T^* with the two-scale momentum theory (Nishino & Dunstan 2020, J. Fluid Mech. 894, A2) to demonstrate that the model can be used to give fast and accurate predictions of large wind farm performance under various mesoscale atmospheric conditions. This new approach could be beneficial for improving annual energy production (AEP) calculations and farm optimisation in the future.