Target speech extraction (TSE) typically relies on pre-recorded high-quality enrollment speech, which disrupts user experience and limits feasibility in spontaneous interaction. In this paper, we propose Enroll-on-Wakeup (EoW), a novel framework where the wake-word segment, captured naturally during human-machine interaction, is automatically utilized as the enrollment reference. This eliminates the need for pre-collected speech to enable a seamless experience. We perform the first systematic study of EoW-TSE, evaluating advanced discriminative and generative models under real diverse acoustic conditions. Given the short and noisy nature of wake-word segments, we investigate enrollment augmentation using LLM-based TTS. Results show that while current TSE models face performance degradation in EoW-TSE, TTS-based assistance significantly enhances the listening experience, though gaps remain in speech recognition accuracy.