My 1996 model is developing the same problem, with 82000 km on. The problem seems to be in the back at about 70 to 80 km/h. I had all the wheels balanced and one mag wheel was sent in to be straightened. But the vibration is still there.
Yes we do use the word shimmy, and it would probably be a good word to describe the wheel wobble problem you have, but for some reason the word is never used in that context.
Raise each front wheel up individually using the jack so its free spinning and grab it with both hands and feel for any play (slackness).
The problem is most likely from your Wishbones.(Wishbones because they resemble the breast bone of a chicken which, English kids used to (maybe still do) pull between there little fingers in a wish making tradition) you maybe call them A-Frames, Y-Struts, swing arms or something like that. They holds the bottom of the suspension strut the most likely place for the slackness is around the back Bush (rubber thing) where the wishbone meets the chasis. The bushes cannot be replaced individually as they are integral to the wishbone. Check here 90% certain this is problem.
Anyway jack the wheel up and check around you should find the source of the problem.
I assume you've tried putting the wheels back to where they were originally. Had this problem on another car a while back and it was a tyre that had gone out of shape. You could see the tread pattern do a dog-leg if you looked around the circumference of the tyre. New tyres cured it