Potentiometric multisensor systems were shown to be very promising tools for the quantification of numerous analytes in complex radioactive samples deriving from spent nuclear fuel reprocessing. Traditional multivariate calibration for these multisensor systems is performed with partial least squares regression—an intrinsically linear regression method that can provide suboptimal results when handling potentiometric signals from very complex multi-component samples. In this work, a thorough investigation was performed on the performance of a multisensor system in combination with non-linear multivariate regression models for the quantification of analytes in the PUREX (Plutonium–URanium EXtraction) process. The multisensor system was composed of 17 cross-sensitive potentiometric sensors with plasticized polymeric membranes containing different lipophilic ligands capable of heavy metals, lanthanides, and actinides binding. Regression algorithms such as support vector machines (SVM), random forest (RF), and kernel-regularized least squares (KRLS) were tested and compared to the traditional partial least squares (PLS) method in the simultaneous quantification of the following elements in aqueous phase samples of the PUREX process: U, La, Ce, Sm, Zr, Mo, Zn, Ru, Fe, Ca, Am, and Cm. It was shown that non-linear methods outperformed PLS for most of the analytes.