My thanks to Roger Crier of Birmingham, England (home of BSA Motorbikes and a lot of other great engineering) for reminding me of the Gyro Hawk.
A video and "info pak" featuring this machine were advertised in several magazines in the 1990s. In fact, I ordered the video and still have it somewhere. Along with the video came a copy of the 1967 article in Science and Mechanics about the Gyro-X. The video itself was not terribly impressive. I haven't watched it in a while but as I recall it was basically a machine--looking like the one pictured in the ad--making slow turns in a parking lot. No thrilling chase shots of a highway-ready vehicle. I didn't resent paying for thing as it clued me in to the Gyro-X. Over the years, a number of people have asked for more info on the Gryo Hawk but I don't have any. And as people have pointed out, the Roadhawk company is not reachable at the address listed. (There was phone number in the ad but I have clipped that to avoid bothering whoever owns that number now.)
If you can shed any light, please leave a comment. I have had an intriguing email from someone who may have spotted a Gyro Hawk and has promised to send pictures the next time they see it. Fingers crossed.