Things are really starting to come together on the Schwinn New World. After some research and questions, I found that the standard Bell cable repair kit will work for the old type Schwinn levers and "Schwinn Built" caliper brakes. That's great news because the Bell kits are cheap, plentiful, but still very effective. The black housing even matches my black New World. I will grant that it isn't 100% period correct, but it matches nicely and will provide peace of mind when I need to hit the brakes.
I then fitted the components, sized the cables and housings, and set up the brakes. My cable routing was a simple, direct type with the front brake to the left and the rear to the right. I like this lay out because it is familiar to me, and it places the important front brake in my dominant left hand. More on the subject can be read at Sheldon Brown's excellent website:
The bicycle is NOT done yet, but is in a state now where I can test ride it. The only remaining parts to go over are the pedals and the saddle. I presently have an original 1940s era saddle on the bike, but it's somewhat beat up and may not last under much riding. I have the makings of another 1940s era original saddle, and I may end up re-upholstering and recovering that saddle with a nice pad and high quality black leather. For the time being, I'm content to test the bike with this relic saddle, which is period acceptable. I may ultimately opt to build my own 1940s saddle though.
The pedals on the bike now are Torrington #8 pedals, which are also period correct to the 1940s. However, they need to be re-gearsed and cleaned up. I actually have a nice set of Torrington #10 pedals I plan on using, which should be good upgrade. Those too are period correct.
I got to take a very short test ride tonight. It was bitterly cold and the wind from the moving bike did not help. That said, I began to get a flavor of the riding characteristics of the bike. It is a weird amalgam of balloon tire and lightweight characteristics. The drive train is like a hybrid of a balloon tire cruiser and a light weight (skip tooth single speed, but with a free wheel and not a coaster brake). The frame rides pretty stiff, but has relaxed frame angles, so again like a mixture of cruiser and light weight. My initial assumption is that it would handle like a slack frame English roadster, but it turns out it's apparently something weird and different entirely. It's just... unique. I only rode down the block and back, so I have more learning to do for this one.