you sure your pressing ~ ?
you running any mods that may have disabled it (or possible edits to the .ini)?
Please correct me if I'm wrong, but from what I have experienced, The Elder Scrolls games don't seem to go well with certain Windows operating systems. I have the same issue with both Morrowind and Oblivion, except on my old Vista.
What about running the game in Windows XP compatibility mode? I'm not sure if it would do anything, but it could be worth a try?
http://www.uesp.net/wiki/Oblivion:Console
Look in the Note section under General Information.
You haven't touched the.ini?
That might be the problem. As I recall, I think you have to make a change to the .ini to set "enable console" from 0 to 1