Terrifying wobble under braking

[gbb]

I should get round to taking the thing apart and regreasing it anyway. So far I've only got as far as servicing the hubs and installing some new brake cables. The headset doesn't feel loose...

I'm suffering the same....looked at my headset among many other things, not an ounce of play detectable.
Brake calipers tight ? Is it possible the brakes are flexing?

 

[benbrangwyn]

I've cured shimmy by grabbing the top tube (which I know you don't have) between my knees. It was during a descent of the Tourmalet in 2015's Etape, and it was sleeting at the top and just about every rider was shivering so badly on the way down the other side and tons of them were shimmying like crazy. You could still apply a shin to the tubing... There was another post I read just now where some guy switched forks on his vintage ride and the bike's handling changed from shimmytastic to solid.