Access is enabled and it was controlling the post-type. I disabled it (for the post type), but it made no difference. I have looked though the permissions and can't see anything wrong.
The site was built with a few test user accounts. These are the ones which show up as 'author' options. The site has now had about 2,100 accounts migrated in from an old Joomla site - none of these are visible in the 'authors' option.
I don't know enough about WP to understand what could be different about the two sets of users.
If it would help, I can give you access to the development site.