That looks like an interesting show, but, alas, it is not the one I'm looking for.
Oy! The name came back to me! It's
The Secret Life of Machines, a BBC production from 1988, 1991 and 1993. The theme song was "The Russians Are Coming."
Here's a YouTube Clip:
[youtube=425,350]http://youtube.com/watch?v=3ehVVpY6XE4
Thanks for trying, guys! I've been trying to remember the name of this show for days, so naturally it comes back to me after I ask you all.