More regarding Steve’s #2 comment above:
I learned a lot about scuttle shake when I disassembled my car prior to sending it off for restoration. I learned even more when I did some of the structural welding. When the chassis is sitting bare on the floor, you can really see where scuttle shake comes from. The scuttle is actually the front bulkhead, which is made up of lots of pressings, triangulations, braces, hinge pillars, etc. The bulkhead is connected to the main frame sections and sills in only a few spots, and over time those spots (and of course any of the other key components like sills) can rust and weaken. The rigidity of the bulkhead is lost, and then things like out-of-round wheels or potholes in the road telegraph through the structure and make the scuttle wobble. Again, when you see the complete structural system before you, it is pretty clear where scuttle shake comes from.
In my case, a complete professional restoration resolved scuttle shake on my car. It is very tight, which is not a term I ever thought I would use in reference to a Healey. It is good to know from Steve’s comments that there are remedies short of a complete restoration. But I imagine Steve’s car was already in good condition. The entire system needs to be extremely sound or you will get scuttle shake. And yes, get good wheels from Hendrix!