Combining kinetic Monte Carlo and static simulations, we follow at the atomic scale the growth of Co nanoislands on Cu(111) surface over a wide range of surface temperatures (130 - 300 K). Atomistic processes responsible for the interlayer mass transport of Co atoms and the formation of 2 ML high nanoislands at tempeatures > 200 K are revealed. Transition from the two- to three-dimensional growth mode with decreasing the temperature is demonstrated. Strain relaxations induced in the Cu substrate and in the Co nanoislands are found to have a strong impact on the formation of triangle Co islands at room temperatures. Results of our theoretical studies are supported by the scanning tunneling microscope measurements.