The surface patterns formed during submonolayer Cu/Ni and Pt/Ni heteroepitaxy upon a Ni(100) substrate have been investigated by kinetic Monte Carlo (KMC) simulations. The two-dimensional (2D) KMC simulations are based upon rate constants for a complete nearest-neighbor set of 729 uncorrelated Cu or Pt atoms and/or Ni site-to-site hopping mobilities. The rate constant activation energies are determined by classical-potential total-energy calculations using an embedded-atom method potential function from the literature. We find that diffusion of Cu atoms occurs at a faster rate and Pt atoms at a slower rate than that of Ni atoms on the flat Ni(100) surface, and the initial nucleation and growth patterns of 2D islands and the kinetic versus thermodynamic control of the growth vary as a consequence. In the temperature and deposition time regime in which we work, the Cu/Ni systems show less than random mixing, while the Pt/Ni systems show more than random mixing. The Cu/Ni system has bonding energies that result in a tendency to segregate toward subdomains of pure Ni and Cu, though kinetic effects in the epitaxy trap the development of the system at small subdomain sizes. The Pt/Ni system has bonding energies giving a tendency to intermix completely, while epitaxial kinetic effects modestly interfere with the complete mixing. The kinetically determined island morphologies under various Cu/Ni and Pt/Ni compositions and deposition rates differ substantially over time periods that are long on the deposition time scale, and therefore the island patterns can become frozen in place.