I wish that I had never seen this thread. I still have my 1980s conventionally shaped bike, which I know I can easily ride with no hands, if I sit up straight and go at a reasonable speed. A little while ago I bought a Raleigh Shopper, just to use for short shopping trips and with it's small wheels it's always seemed a bit wobbly. Despite this, since reading this thread, I got to thinking 'I wonder if ...?'.
I'm so glad that I spotted this on a Sheldon Brown page:
The stock Raleigh Twenty headset uses the same bottom section as other Nottingham 3-speeds, with 25 5/32" balls, but the upper headset uses a nylon bushing.The extra friction of this bushing plus the geometry of the stock fork generally make it impossible to ride a stock Twenty no-hands. Replacing the upper headset solves this problem.
I'm not even going to try it; I've had one serious injury in a bike accident, I don't want another.