The spin-up from rest of an isothermal liquid metal driven by a rotating magnetic field is examined by means of numerical simulations of the axisymmetric Navier-Stokes equations. The geometry studied is a cylindrical cavity with an upper free surface and the aspect ratio equal to unity. The magnetic Taylor numbers comprise the range 10³ to 10⁶. We identify three different flow regimes including a Stokes regime, a non-linear and an oscillatory regime, which occur in dependence on the time and the magnetic Taylor number. We discuss the flow structure and determine the characteristic times required for the flow to be fully developed.