Our recently developed linearized augmented-plane-wave thin-film method is used to determine the electronic structure and magnetism of Ni overlayers on Cu(001). Accurate ab initio self-consistent spin-polarized semirelativistic band calculations are reported for (i) a clean five-layer Cu(001) slab and (ii) the same Cu slab plus one or two $p(1\ifmmode\times\else\texttimes\fi{}1)$ layers of Ni on either side. Results presented include charge and spin densities, work function, band structures, projected density of states, magnetic moments, and direct and transferred hyperfine fields. Both surface and interface effects are found to be important. The Ni overlayers are not magnetically dead: The Ni layer adjacent to the Cu has its moment decreased from the bulk value to 0.39${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ for the single Ni overlayer and to 0.47${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ for the two-Ni-thick layers; the surface Ni layer in the two-layer Ni on Cu film has its moment increased somewhat to 0.68${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$. This reduction in moment for the interface Ni arises primarily from charge transfer onto Ni sites from the Cu substrate. By contrast, the increase in moment of the Ni surface atoms arises in large part due to the dehybridization of the $p$ electrons from the $d$-band electrons; these $p$ electrons become more delocalized and spill out into the vacuum region. A similar effect was also observed for an unsupported Ni monolayer. In the case of the Ni monolayer on Cu, the total number of Ni electrons is almost the same as for bulk Ni. Here, the loss of electrons due to the dehybridization of $p$ electrons is nearly canceled by the increase from its interface with the Cu substrate; the decrease in magnetic moment (to 0.39${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$) agrees with electron-capture experiments.