(Before you ask I checked the forum here and it was inconclusive)

In the New Nightmare the CG cutscenes and intros won't show, suggestions?

Also I am using Windows 7.

I tried it in Windoes 7 32 and I had no problem.
You shouldn't have any codecs problem as the videos are in bink format.

I think I had the same issues. Have you tried messing with the setup that GOG has included. I don't have the game installed anymore, but I can try to see what I can do to help.

I have Win 7 64bit.

I found the forum post that I got help in, but seen you posted there as well. Using Win Xp sp3 and run as admin was the only thing that helped me as well as Alt-tabbing out.
Hm, not sure if downloading Bink would help, but here is the link for it.
http://www.radgametools.com/bnkdown.htm

I'm not sure what other advice that I can give other than messing with compatibility mode. There's nothing else I could google up that would help. If I can find more help, I'll update this post!

EDIT: I've found a safe site for an updated UK/US launcher. I shall PM you the site cause of the contents on it and I don't want to get in trouble :P