The infamous nylon headset bushing. (For a general description of the headset and a quick service, see this post). Instead of a top race, this is what the Raleigh Twenty uses for a bearing at the top of the head tube. The bottom end has ball bearings - there are photos of that below.
This is the rear view of the nylon bushing. Note the cutout in the steerer tube that allows the steerer clamp to grip the stem. (Photos of the original headset configuration are here).
Here, the nylon bushing is being taken out. It's all really very curious. I wonder why Raleigh did it that way?
View of the cup at the other bottom end of the headtube. All greased and ready to be assembled.
New balls please! This is a photo of the balls on the crown race, all greased and ready for assembly. They are new shiny ones, 25 of 5/32" size.
I'm going to use the top half of this Madison M:Part Sport threadless headset. That will replace the nylon bushing at the top of the head tube. At the bottom of the head tube, I'll leave the original cup and bearing arrangement, because it's absolutely fine.
Just one spacer above the bearing cap, then the original lock nuts, turned carefully to apply the right amount of pressure - so that the forks rotate freely with no tightness or binding, and no play in the headset either.
Here goes the new stem. It's an alloy quill with an adjustable alloy stem with three bolt handlebar clamp. Madison M:part again. Titec Hell Bent alloy bars, with a small rise. 
The bars were too wide, so I had to saw the ends off (to ~56cm total width). Simply filed the ends off after cutting with a junior hacksaw.
Ta Da! The bike, now even more transformed to the 21st century. The difference that the headset makes to the ride quality is amazing. It's now really responsive, but at the same time stable. I can ride hands off. Previously, you had to apply positive force to the bars to turn. Now you can "think" and lean the turns, just like a "proper" bike! In fact, it feels MUCH better to ride now than my commuting Dahon Speed D7 folder. The Triumph Twenty is a small bike with a big bike's personality!
View from the driving seat. Tioga Power Stud bar ends, give a welcome change in hand position.
Trigger shifter is in easy thumb and index finger reach (as is the bell!). Swanky new, but oh so cheap, brake levers. Brakes are really good now, even the back one is not bad, despite the original sidepull caliper and pads. They will improve further when I replace the steel rim with an alloy one.
Looking good. The main work left now is to replace the rear steel rim with an alloy one, just like the front wheel. I'm still musing over the mudguards. Should they stay or should they go?
Proudly sporting the Triumph head badge. Actually, that is the main reason that I didn't want to send it off for re-spraying or powder coating. I couldn't figure out how to take off the head badge without damaging it. It seems to be riveted in place (three rivets I think). Awkard to reach the back of the rivets inside the head tube.
Very satisfying to reach this far, with no hitches along the way. She really is a lovely bike to ride now. UPDATE: 22 September 2010
Days after the above blog post, she moved to a different town to be used as a commuter by a student cousin of mine. So I won't get a chance to replace that rear steel rim for a while. Meanwhile, my 8 year old son has implored me to buy another one and to keep the old steel bars (because they were "cool" and "retro" - he's EIGHT for goodness' sake!). By all accounts, in virtually daily use, she has been completely reliable thus far.
