A constructive algorithm using Chebyshev spectral collocation is proposed for computing trustworthy approximate solutions of linear and weakly nonlinear delayed partial differential equations or initial boundary value problems, with continuous and bounded coefficients. The boundary conditions are assumed to be Dirichlet. The solution of linear problems is obtained at Chebyshev grid points in space and a given interval of time. The algorithm is then extended to systems with weak nonlinearities using perturbation series, which yields nonhomogeneous initial boundary value problems without delay. The proposed methodology is illustrated using examples of linear and weakly nonlinear heat and wave equations with bounded continuous space-time varying coefficients.