Do you remember what time of day you used to see the van trains?
I have a 1965 Freight Working Timetable for the Birkenhead to Chester line (but it doesn't cover the West Kirby line). There was a steady trickle of goods trains booked from the Birkenhead area through the day (and night), but none mention Moreton as the origin or destination.
Most seemed to go to fairly local yards like Warrington, Saltney (Chester) or Crewe Basford Hall, where presumably the wagons would be re-marshalled for other destinations. A couple of trains in 1965 went through to further destinations, like Carlisle, York or Banbury, though these originated at places like Birkenhead Shore Road Goods depot or Morpeth Dock, so presumably not the Cadbury trains.
This post on Facebook shows one of the Cadbury van trains leaving Moreton in August 1966 behind Birkenhead shed's Crab 2-6-0 no. 42727.
As it says in the caption, many of the vans were dedicated to Cadbury's traffic and stenciled "Return to Cadburys Moreton Wirral".