King Street is mapped https://www.openstreetmap.org/way/61095857  agreeing with OS OpenMap Local (April 2024)
Church Street West https://www.openstreetmap.org/way/61047130 joining Church Street East https://www.openstreetmap.org/way/68336678 also agrees with OS OpenMap Local but the detail is not good enough to confirm the name by the note but I don;t think it is King Street