The aim of this study was to develop and verify a hybrid method for modelling machining errors in a specialized machine tool used for reconditioning and profiling train wheels, combining the finite element method (FEM) with artificial neural networks (ANNs). A detailed numerical model was created, incorporating both machine tool components and the machined wheelset. This model enabled analysis of cutting-force-induced elastic deformations under realistic boundary conditions, variable system stiffness, and the specific method of supporting the wheelset in centres and on driving rollers. Results from FEM simulations were used to build a database for training an ANN capable of rapidly and accurately predicting radial and axial deformations as functions of cutting forces and related machining parameters. This allowed the development of a functional form of machining error suitable for use in numerical control (NC) systems. Testing confirmed that the ANN effectively reproduced the error magnitudes obtained from FEM simulations. Its application significantly improved prediction efficiency while keeping computational time very low. The proposed method supports indirect error compensation through real-time measurement of cutting forces and corresponding tool-path correction, representing an important step toward intelligent machining systems. Further work on refining the functional form of machining errors will focus on expanding ANN inputs to include wheelset design parameters, enhancing the model’s generality and compensation capabilities.