Just spent an hour messing about with this game rinstalling drivers and tweaking settings to get it working so sharing the fix here.

Symtoms:
When starting the game the intro video is messed up; It's squashed out of shape. The sound is fine though, the game is actually running but you can't do anything with it because you can't see. I had to Alt-Tab out to the desktop to quit.

The fix:
Our PC display is a strange resolution, 1440x900. The game seems unable to autodetect that and tries an inappropriate setting. You need to manually edit the following file:
%APPDATA%\DisneyInteractiveStudios\LEGOPirates\pcconfig.txt

As the name suggests it's a text file so Notepad will do. Change these four lines near the top:
ScreenWidth 1280
ScreenHeight 800
WindowWidth 1280
WindowHeight 800

Change both width & height setting to match your display, in my case that is 1440 and 900 respectively. The game (and my children) are perfectly happy with this and it just works.