firstly I would double check the function which is creating your custom post type,
within that function there should be an element called: rewrite
ie:
register_post_type( 'products',
'menu_position' => 25, // below pages
'public' => true,
'show_ui' => true,
'rewrite' => array( 'slug' => 'product' ) <-- this is what you need!
);
also check the register_taxonomy function for the same!
ie:
register_taxonomy(
'team',array('product_types'),
array(
'public' => true,
'show_ui' => true,
'show_in_nav_menus' => true,
'query_var' => true,
'hierarchical' => true, <-- this is needed!
'rewrite' => true <-- this is what you need!
));
the only thing left to check is:
your permalink structure is set to /%postname%/
you may have to reset to default, save it,
then re-set to /%postname%/ and save,
hope that helps :)
Marty