Hand hygiene is among the most effective daily practices for preventing infectious diseases such as influenza, malaria, and skin infections. While professional guidelines emphasize proper handwashing to reduce the risk of viral infections, surveys reveal that adherence to these recommendations remains low. To address this gap, we propose UWash, a wearable solution leveraging smartwatches to evaluate handwashing procedures, aiming to raise awareness and cultivate high-quality handwashing habits. We frame the task of handwashing assessment as an action segmentation problem, similar to those in computer vision, and introduce a simple yet efficient two-stream UNet-like network to achieve this goal. Experiments involving 51 subjects demonstrate that UWash achieves 92.27% accuracy in handwashing gesture recognition, an error of <0.5 seconds in onset/offset detection, and an error of <5 points in gesture scoring under user-dependent settings. The system also performs robustly in user-independent and user-independent-location-independent evaluations. Remarkably, UWash maintains high performance in real-world tests, including evaluations with 10 random passersby at a hospital 9 months later and 10 passersby in an in-the-wild test conducted 2 years later. UWash is the first system to score handwashing quality based on gesture sequences, offering actionable guidance for improving daily hand hygiene. The code and dataset are publicly available at https://github.com/aiotgroup/UWash