The spin reorientation transition in fcc Co/Ni/Cu(001) epitaxial ultrathin films as a function of Co and Ni film thickness is studied by the combination of photoelectron emission microscopy and x-ray magnetic-circular-dichroism spectroscopy at the Ni L_2,3 edge. This microspectroscopic technique allows one to extract local quantitative information about the Ni magnetic properties on a submicrometer scale. Domain images in the thickness range of 1.4-2.6 atomic monolayers (ML) Co and 11-14 ML Ni show that the spin reorientation occurs as a function of both Co and Ni thicknesses. Increasing the Co thickness or decreasing the Ni thickness leads to a switching of the magnetic easy axis from [001] out-of-plane to <110> in-plane directions. A constant effective Ni spin moment similar to the bulk magnetic moment is observed. The Ni orbital to spin moment ratio shows distinctly different values for out-of-plane magnetization (0.080±0.005) and in-plane magnetization (0.053±0.005). This is discussed in terms of the connection to the Ni magnetocrystalline anisotropy. The domain density of the perpendicular magnetization increases towards the spin reorientation transition line.