"The nuclear explosion would only impart say, a 100-gees, to the reaction balls during detonation. This gee-load during detonation is what imparts the reaction ball with an instantaneous velocity in a direction perpendicular to its original plane of rotation (3rd dimension thrust). The purpose of the long tether is to allow the reaction balls to swing forward through a long gradual arc, using it like a pendulum to store the energy of the bomb. I don't think the tether would experience the same extreme gee-loading during this process, only the reaction ball, but I could be wrong. A pendulum with a steady force applied, like gravity, is relatively straightforward calculus. But when I add the additional spinning condition instead of gravity, it creates a non-steady centripetal force that depends on radius and initial rotation frequency. I find it quite confusing in fact, and I haven't had a chance to sit down and work it out."

That may be, but I was trying to start with the simple. In other words, the cable will need provide force [provide acceleration to the vehicle].
So if the vehicle weighed 10 tons. The cable weighed 10 ton. If the most acceleration provided to vehicle was 4 gees [39.2 m/s/s]. Then the strength of cable would need to support 20 tons times 4 (be able to lift 80 tons).

"I don't know off the top of my head how much energy is released by a nuclear bomb. How small can they be made? This system doesnt necessarily have to use the Czar bomb to propel it. Another interesting aspect of this is that the pendulum effect could be extremely efficient means of storing the energy of the bomb and converting it into kinetic energy. The Orion pusher-plate system used shock absorbers, which would instead waste huge amounts of energy by converting the impulse of the explosions into heat! The Nuclear Pulse AutoGyro would act like a pure spring instead, but with a very gradual vibrational motion in the direction of travel. "

Google suitcase nuke:
"The idea of portable nuclear devices was not a new one.

In the 1950s and 1960s, the U.S. made the first ones, known as the Special Atomic Demolition Munition. It was a "backpack nuke" that could be used to blow up dams, tunnels or bridges. While one person could lug it on his back, it had to be placed by a two-man team."
http://www.rawstory.com/news/mochila/Suitcase_nukes_said_unlikely_to_exi_11102007.html

Wiki, Special Atomic Demolition Munition:
"In the 1950s and 1960s, the United States developed several different types of lightweight nuclear devices. The main one was the W54, a cylinder 40 by 60 cm that weighed 68 kg. It was fired by a mechanical timer and had a variable yield equivalent to between 10 tons and 1 kiloton of TNT. 300 SADMs were assembled and remained in the US arsenal until 1989."

So this thing gives the energy of 1000 tonnes of TNT and weighs 68 kg, not particularly efficient in terms nuclear weapons but it's pretty small. Probably want something a couple times bigger.
Though I suppose what you might want is Hydrogen bombs rather than atomic bombs. Some other info here:
http://www.tinyvital.com/Misc/nukes.htm