I have got a Wordpress structure such as this.

HOME
HOME > BLOGS
HOME > BLOGS > BLOGNAME1 > SINGLE PAGES
HOME > BLOGS > BLOGNAME2 > SINGLE PAGES
HOME > BLOGS > BLOGNAME3 > SINGLE PAGES

I would like the BLOGNAME1, BLOGNAME2, BLOGNAME3 category pages to make use of exactly the same template.

From what I will tell, I am after something to set up functions.php that states if category is really a descendant from the BLOGS category, its category page should make use of a particular template.

You will find some similar discussions here but they'd all pressure these subcategories to inherit the BLOGS template, which is not what I am after.

Any ideas?

The very first that found mind is by using the next fork inside archive.php (or index.php without having archive.php)

<?php if(is_category(array('BLOGNAME1', 'BLOGNAME2', 'BLOGNAME3'))) : ?>

 the custom template HTML goes here

<?php else : ?>

 normall flow

<?php endif; ?>

You should use category ID, slug or/and title as array values. You may also mix them. Find out more here: http://codex.wordpress.org/Function_Reference/is_category