We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general L-p(W-1,W-p) setting. For p = 2, optimal convergence rates in the L-2(H-1) and C-0(L-2) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.