We present Monte Carlo simulations of the first stages of the coherent precipitation of Cu in α-Fe during thermal ageing and under irradiation. The simulations are based on a diffusion model by vacancy and self-interstitial jumps, the parameters of which are fitted on ab initio calculations. The simulations of precipitation kinetics during thermal ageing are compared with experimental ones. They reveal that precipitates containing up to several tens of atoms can be much more mobile than individual copper atoms, due to the trapping of vacancies in Cu. Monte Carlo simulations are also used to analyze the coupling between fluxes of point defects and Cu which occurs under irradiation and the possible resulting radiation induced segregation phenomena.