Predicting the mechanical properties of multiscale nanocomposites requires simulations that are costly from a practical viewpoint and time consuming. The use of algorithms for property prediction can reduce the extensive experimental work, saving time and costs. To assess this, ternary poly(hydroxybutyrate-co-hydroxyvalerate) (PHBV)-based bionanocomposites reinforced with graphene oxide (GO) and montmorillonite nanoclay were prepared herein via an environmentally friendly electrochemical process followed by solution casting. The aim was to evaluate the effectiveness of different Machine Learning (ML) models, namely Artificial Neural Network (ANN), Decision Tree (DT), and Support Vector Machine (SVM), in predicting their mechanical properties. The algorithms’ input data were the Young’s modulus, tensile strength, and elongation at break for various concentrations of the nanofillers (GO and nanoclay). The correlation coefficient (R2), mean absolute error (MAE), and mean square error (MSE) were used as statistical indicators to assess the performance of the models. The results demonstrated that ANN and SVM are useful for estimating the Young’s modulus and elongation at break, with MSE values in the range of 0.64–1.0% and 0.14–0.28%, respectively. On the other hand, DT was more suitable for predicting the tensile strength, with the indicated error in the range of 0.02–9.11%. This study paves the way for the application of ML models as confident tools for predicting the mechanical properties of polymeric nanocomposites reinforced with different types of nanofiller, with a view to using them in practical applications such as biomedicine.