You will need some specialist recovery software. We were recommened a program called ZAR (Zero Assumption Recovery). Other similar programs are available, but ZAR worked well for us and is free for this purpose.
Install ZAR, then insert the damaged SD card into your PC's card reader. Launch ZAR. (You may have to disable your PC's security software, in which case we recommend you also disable the internet connection).
When prompted, click the Image Recovery (Free) option. This will help you to recover photos from an SD card.
ZAR will now look for devices that are installed, and you need to point it toward the offending SD card. Select the correct disk and click Next. ZAR will now analyse your SD card. This will take a few minutes.
Once this is complete you'll see a list of the recovered files. On most SD cards the chances are that clicking the 'Root' checkbox will select all the images. Click Next and you can then select the folder into which recovered files will be placed. By default this will be the same as the folder your chose to save (so 'Root', in all likelihood).
One important point: you will be asked to select a destination folder for the recovered files. Make sure you choose one on your PC's hard disk and not back on to the corrupt SD card. The latter option will help you not one jot!
It will take a few minutes for the files to copy across. Once this is complete browse to the destination folder you selected. You should find most if not all of the files you wanted to rescue.