So I got this working by adding /tag or /category for the rewrite slug, rather then publications or publications/%publications_tag%. I initally thought both taxonomy types could live in the parent post-type and wordpress would figure the rest out.
Tags:
'rewrite' => array(
'slug' => 'publication/tag',
'with_front' => false ),
Looks like: domain.test/publications/tag/example-tag/ and use’s the axonomy-publications_tag.php file.
Category:
'rewrite' => array(
'slug' => 'publication/category',
'with_front' => false ),
Looks like: domain.test/publications/category/example-category/ and use’s the axonomy-publications_category.php file.