Mr G I have EXACTLY the same problem.
A while back I got a set of second hand rims with tyres on them , had them put on and went for a drive. The result was bloody horrible. The car was shaking so badly that I went and got the wheels balanced, which made it just a bit more bearable. After the wheel balancing, it was ok for under 80k driving, but above that it was shaking so bad that I thought the car was going to break apart. I thought it might have been the rims being out of shape, but anotheer tyre shop guy said to drive it for a bit longer and see how it goes. Gradually, it got better, now its good at 100 and slightly shakes above 140k. I was going to get a wheel alignment the other day as the tyre is wearing out on the inside but they couldn't get the car up and referred me to another place. I haven't been yet, but I'll I hope this helps. I'll let you knw what happens when I get the wheels aligned.
Before the current skyline, I had a series one R33. and a wheel alignment fixed the shaking problem. So I'm counting on it again.