Thermal errors in machine tools significantly impact machining precision and productivity. Traditional thermal error correction/compensation methods rely on measured temperature-deformation fields or on transfer functions. Most existing data-driven compensation strategies employ neural networks (NNs) to directly predict thermal errors or specific compensation values. While effective, these approaches are tightly bound to particular error types, spatial locations, or machine configurations, limiting their generality and adaptability. In this work, we introduce a novel paradigm in which NNs are trained to predict high-fidelity temperature and heat flux fields within the machine tool. The proposed framework enables subsequent computation and correction of a wide range of error types using modular, swappable downstream components. The NN is trained using data obtained with the finite element method under varying initial conditions and incorporates a correlation-based selection strategy that identifies the most informative measurement points, minimising hardware requirements during inference. We further benchmark state-of-the-art time-series NN architectures, namely Recurrent NN, Gated Recurrent Unit, Long-Short Term Memory (LSTM), Bidirectional LSTM, Transformer, and Temporal Convolutional Network, by training both specialised models, tailored for specific initial conditions, and general models, capable of extrapolating to unseen scenarios. The results show accurate and cost-effective predictions of temperature and heat flux fields, which lay the groundwork for flexible and generic thermal error correction through a shift from direct error prediction to spatio-temporal thermal field modelling.