A novel, yet generic, Bayesian approach to parameter inference in a stochastic, spatio-temporal model of dispersal and colonisation is developed and applied to the invasion of a region by an alien plant species. The method requires species distribution data from multiple time points, and accounts for temporal uncertainty in colonisation times inherent in such data. Covariates, such as climate parameters, altitude and land use, which capture variation in the suitability of sites for plant colonisation, are easily incorporated into the model. The method assumes no local extinction of occupied sites and thus is primarily applicable to modelling distribution data at relatively coarse spatial resolutions of plant species whose range is expanding over time. The implementation of the model and inference algorithm are illustrated through application to British floristic atlas data for the widespread alien Heracleum mantegazzianum (giant hogweed) assessed at a 10 × 10 km resolution in 1970 and 2000. We infer key characteristics of this species, predict its future spread, and use the resulting fitted model to inform a simulation-based assessment of the methodology. Simulated distribution data are used to validate the inference algorithm. Our results suggest that the accuracy of inference is not sensitive to the number of distribution time points, requiring only that there are at least two points in time when distributions are mapped. We demonstrate the utility of the modelling approach by making future forecasts and historic hindcasts of the distribution of giant hogweed in Great Britain. Giant hogweed is one of the worst alien plants in Britain and has rapidly increased its range since 1970, yet we highlight that a further 20% of land area remains susceptible to colonisation by this species. We use the robustness of this case study to discuss the potential for modelling distribution data for other species and at different spatial scales.