The integration of high-precision cellular localization and machine learning (ML) is considered a cornerstone technique in future cellular navigation systems, offering unparalleled accuracy and functionality. This study focuses on localization based on uplink channel measurements in a fifth-generation (5G) new radio (NR) system. An attention-aided ML-based single-snapshot localization pipeline is presented, which consists of several cascaded blocks, namely a signal processing block, an attention-aided block, and an uncertainty estimation block. Specifically, the signal processing block generates an impulse response beam matrix for all beams. The attention-aided block trains on the channel impulse responses using an attention-aided network, which captures the correlation between impulse responses for different beams. The uncertainty estimation block predicts the probability density function of the UE position, thereby also indicating the confidence level of the localization result. Two representative uncertainty estimation techniques, the negative log-likelihood and the regression-by-classification techniques, are applied and compared. Furthermore, for dynamic measurements with multiple snapshots available, we combine the proposed pipeline with a Kalman filter to enhance localization accuracy. To evaluate our approach, we extract channel impulse responses for different beams from a commercial base station. The outdoor measurement campaign covers Line-of-Sight (LoS), Non-Line-of-Sight (NLoS), and a mix of LoS and NLoS scenarios. The results show that sub-meter localization accuracy can be achieved.