Ok if anybody else is having the same problem I have figured out a workaround.
The shortcode doesn't work with WP 3.9.1 full stop, but the widget does!
Widgets only work in sidebars but I don't use sidebars on my site, so below you will find a way that will allow you to put the widget anywhere on your site that you choose.
Basically you just have to register a new sidebar in your functions.php file, drop the widget in your new sidebar & configure it, create a new page template that loads the widget then create a new page.
Step by Step :
1) Open functions.php - find the following function - function twentytwelve_widgets_init() (this will be slightly different depending on which theme you are using - i'm using TwentyTwelve) and add the following code right before the closing curly bracket :
'before_widget' => '',
'after_widget' => '',
'before_title' => '<h4>',
'after_title' => '</h4>',
2) Go to Appearance > Widgets. Then drag and drop the Radio Station: Upcoming DJ On-Air widget into your new sidebar "mysidebar". Configure it how you wish and save it.
3) Create a new page template file (if you don't know how to do this click here) and then add the following code where you wish the widget to appear :
<div><?php dynamic_sidebar( 'mysidebar' ); ?></div>
4) Create a new Page and assign the new page template that you created in Step 3 to it from the dropdown menu on the right.
Obviously you can just drop that sidebar code wherever you want, it doesn't have to be done on it's own separate page, but I've just explained it like this to make it easy to understand.
Hope somebody finds it useful.