Reliable radar pulse classification is essential in Electromagnetic Warfare for situational awareness and decision support. Deep Neural Networks have shown strong performance in radar pulse and RF emitter recognition; however, on their own they struggle to efficiently learn new pulses and lack mechanisms for expressing predictive confidence. This paper integrates Uncertainty Quantification with Lifelong Learning to address both challenges. The proposed approach is an Evidential Lifelong Classifier (ELC), which models epistemic uncertainty using evidence theory. ELC is evaluated against a Bayesian Lifelong Classifier (BLC), which quantifies uncertainty through Shannon entropy. Both integrate Learn-Prune-Share to enable continual learning of new pulses and uncertainty-based selective prediction to reject unreliable predictions. ELC and BLC are evaluated on 2 synthetic radar and 3 RF fingerprinting datasets. Selective prediction based on evidential uncertainty improves recall by up to 46% at -20 dB SNR on synthetic radar pulse datasets, highlighting its effectiveness at identifying unreliable predictions in low-SNR conditions compared to BLC. These findings demonstrate that evidential uncertainty offers a strong correlation between confidence and correctness, improving the trustworthiness of ELC by allowing it to express ignorance.