There indeed trains to London from Liverpool Lime Street. One every hour in fact. However, the 14:47 service to Warrington Bank Quay departs at the same time as the 14:47 to London Euston, so they depart at the same time. Announcements are made three minutes before departure, so the announcements for these services clash at the same time, and so only one is done, usually the Warrington train as shown here in the video. You would think they could just have an announcement four minutes earlier for the London train given it's higher priority as a service though wouldn't you?
