Accurate representation of manufacturing-induced variability is essential for reliable prediction of the thermo-mechanical behaviour of engineering components. In this work, we present a Bayesian framework for inferring manufacturing-related random field parameters from temperature measurements acquired at the manufacturing line and from small-scale structural specimens, and for transferring this information to larger, operational-scale products.Temperature distribution during manufacturing is modelled as a spatially distributed random field governed by a stochastic partial differential equation (SPDE), providing a physics-informed prior that reflects underlying heat and mass diffusion. Sparse and noisy temperature measurements are assimilated using Bayesian inference to identify the parameters of the SPDE-based random field, such as characteristic length scales and variance, while accounting for discrepancies between manufacturing-scale and application-scale models. The inferred random field serves as a reduced-order representation of manufacturing uncertainty, enabling efficient propagation of thermal variability to larger structures for which direct measurements are unavailable. Numerical examples demonstrate that temperature variability inferred from manufacturing data and small structures can be consistently applied across scales, leading to improved predictions and reduced uncertainty compared to uncalibrated models. The proposed SPDE-based Bayesian framework provides a link between manufacturing process data and structural-scale simulations, supporting uncertainty-aware modelling and digital twins of manufactured systems.