This paper proposes a homogenization model for transmission lines with nonuniformity manifested in the random permittivity profile of their surrounding medium. The statistics of the permittivity profile are defined in terms of a finite series of correlated random variables corresponding to discrete samples taken along the longitudinal dimension of the line. Principal Component Analysis is employed to reduce the dimensionality of the random space that characterizes the uncertainty of the structure. Next, Polynomial Chaos expansion is used to capture the dependencies of the resulting homogeneous effective permittivity with the reduced-order random parameters that define the variability. Such construction is performed efficiently with the use of a Sparse Grid integration technique. In particular, the statistics of the propagation time of a wave traveling in the uncertain medium are calculated with the proposed homogenization model and validated with Monte Carlo simulations.