Hi,
Yes, you're right. If your site has not been indexed by google yet, you won't see any result when you try to search for your own domain.
However, I just tried:
site:prentonhigh.wirral.sch.uk
There's one entry already - your home page.
So if you select your domain and search for, say, "Prenton High". That one page that has been indexed by google appears.
As more of your sites get indexed by google, you will find more pages appear when you select your domain.
By the way, could I offer a suggestion for your site?
I did a view source of your page and found that there's no Itemid.
To understand why we need Itemid, please take a look at the page:
http://www.kksou.com/php-gtk2/index.php?option=com_googlesearch&Itemid=81
Did you notice what's the difference between your search result page and my search page? Did you notice that your search page is a little bit empty? None of the modules that appear in your home page appears in your search page.
In Joomla, it makes use of the menu id (or Itemid) as a means of deciding which module to appear in what component. So if you want to have precise control of what modules to appear in a component, make sure you create a menu item of the component in the Menu Manager.
Once you have a menu item created in the Menu Manager that links to your googleSearch component, you will be able to select exactly which modules to appear in your search result page. By having some modules appear in the search page, it makes that page feels more consistent with the rest of your Joomla pages.
Regards,
/kksou