Microstructure simulation plays a vital role in the numerical investigation of the process-microstructure-property linkage for additive manufacturing (AM). Among various microstructural modelling strategies, cellular automata (CA) is a widely used method for simulating the AM microstructure which strikes a balance between computational cost and physical relevance. This makes it a more practical choice compared to other alternative methods such as kinetics Monte Carlo and phase-field simulation. Despite its practicality, the computational cost of CA modelling remains a significant obstacle to its application in large-scale simulations or inverse optimisation problems that require numerous iterative simulation runs. To address the challenge of CA computational cost, we developed a novel microstructure simulation framework called neural cellular automata (NCA), which leverages convolutional neural networks to replace the local updating rules in the traditional CA methods. Once trained, NCA can be treated as an accurate but inexpensive surrogate for microstructure modelling that captures essential solidification features, such as preferred growth direction and competitive grain growth. It outperforms the conventional CA method by orders of magnitude in terms of computational speed without sacrificing predicted accuracy. This capability in fast forward evaluations further benefits the inverse calibration of nucleation parameters in the microstructure model. The calibrated NCA model can give accurate predictions of AM microstructures as compared to the experimental observations under various process conditions. Notably, NCA can be more than a black-box machine learning solution but a general simulator that is compatible with various mesh sizes or time increments after training with data from different spatial-temporal discretisations.