This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice---a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja, Kuo, Sloan (SIAM J. Numer. Anal., 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.