One of my followers, PanzerKaput, has done a little investigating himself, and has found Sidney Bradshaw's entry in the Edinburgh Gazette! I didn't even know there was an Edinburgh Gazette. After searching fruitlessly through the London Gazette, there is his name and number at the top left of the page!
Still no mention of a citation for what he did to deserve the medal, though.
Thanks for your help, PK!
I just received the reply from the IWM to my original request. As well as suggesting the National Archives and the London Gazette, they suggested I find the unit diary for the 7th Btn. Border Regt., which is there in the National Archives as well. Unfortunately, it's not digitised, so I can't peruse it from the comfort of my PC in Australia :(. One of my in-laws is living and working in London at the moment, so I'll see if there's any possibility they can get a copy of the citation.
The IWM also suggested getting in touch with the Border Regt. Museum in Carlisle (something young PK also advised). They ask for a minimum GBP 25 donation for their research, which really isn't that bad seeing as they don't get much funding from the MoD or governement in general, but I think I'll try and find out as much as I can before I resort to shelling out the readies (you can tell I'm descended from Scots!).