We study the restructuring of solid nickel catalyst nanoparticles during carbon nanotube growth by environmental transmission electron microscopy and multiscale modeling. Our molecular dynamics/continuum transport calculations of surface-diffusion-mediated restructuring are in quantitative agreement with the experimentally observed catalyst shape evolutions. The restructuring time scale is determined by reduced Ni diffusion through the stepped Ni-C interface region where the catalyst surface strongly anchors to the growing nanotube.