How do you reorder the custom taxonomies on the right side of a custom post edit page?

If you’re talking about the edit screen boxes for these taxonomies, you can do that by changing the $content and priority parameters in add_meta_box()

$context (string) (Optional) The context within the screen where the
boxes should display. Available contexts vary from screen to screen.
Post edit screen contexts include ‘normal’, ‘side’, and ‘advanced’.
Comments screen contexts include ‘normal’ and ‘side’. Menus meta boxes
(accordion sections) all use the ‘side’ context. Global Default value:
‘advanced’

$priority (string) (Optional) The priority within the
context where the boxes should show (‘high’, ‘low’). Default value:
‘default’

add_meta_box( 'my-meta-box', 
 My Meta Box, 
 'my-meta-box-callback-function', 
 null, //unless you need the $screen parameter 
 'advanced', //change this $context parameter for placement
'default', //change this  $priority parameter for order within placement
 null // unless you need $callback_args 
 );