The nucleation kinetics of ordered intermetallic alloy of Cu$_3$Au based on L1$_2$ superlattice are obtained by molecular dynamics simulation utilising Sutton–Chen potential under different pressures. The nucleation mechanism for the system is examined with classical nucleation theory at a slow cooling rate. The crystal-type bonded pairs described by indexes of the Honeycutt–Andersen method are considered as embryos in the system. The critical nucleus radius, the interfacial free energy and the nucleation rate are determined at given temperaturesunder 0, 5, 10, 15 and 20 GPa pressures. The structural properties at different pressures of the system are analysed with radial distribution function. As a result, while the critical radius and the interfacial free energies of the system decrease, the nucleation rates increase with increasing pressure.