Not going to happen. We STILL cant make the necessary length nanotubes, and thats the necessary first step. Getting an earlier start on nanotubes would be nice, but almost certainly you need pretty much todays tech to build sufficiently long nanotubes.
I would guess that with development starting in the 50s we might have sufficently long tubes by today. Possibly even by 2000, say. But figuring out how to do that on an industrial scale would take longer, weaving those fibers into sufficiently strong threads/ribbons, to build thousands of kilomeyers of said ribbon, without a single flaw, thats all going to take time and money. Billions and billions of dollars of money.
We also havent figured out how to practically get a load up and down such a ribbon in a practical and economic fashion, although thats "just" an engineering problem.
So... without an early pod that advances all tech, im pretty sure 2010 is about as early as engineering tests could happen, and i cant imagine a full scale tower by then. Thatd be more like 2020, imo.