A few comments on the space stuff:
1) I'd be dubious of the creation of the "World Space Agency" that soon; all early launch vehicles were modified ICBMs and I frankly can't image the Soviets and US being willing to share that knowledge.
Again, there's no patent on the laws of physics. Nor were the US, Germans, and Russians the only ones researching rocketry. The British were doing significant rocket research of their own in the 1930's; a few more German rocket scientists defecting and/or being captured by the western forces as opposed to the Soviets would boost the British program as well as the US program. British R&D (in cooperation with French R&D in an effort similar to the development of OTL's Concorde) would provide the nucleus for a European-run space program. Likewise, a major industrial power like Japan could start its own program based upon publicly available research.
2) Solar power satellites are overrated. They cost ridiculous amounts of money to build and launch, only for a modest return of power. Building large terrestrial arrays in the desert and piping the energy out would much more simple (but still almost impossible).
Probably the most implausible, given that 1960's solar arrays had about a 9% efficiency rate, but over the long run the effort would be worth it (particularly once petroleum runs out).
3) Who is funding all this massive colonisation effort? Top of my head, you're talking about a $1-3 trillion per annum program. NASA's budget is about $16 billion, and ESA's, about $5 billion. Where is all this incentive to spend massively more than OTL on space?
Using OTL's technology, you'd be right, but even that figure's an exaggeration. This ATL presumes that aerospace technology doesn't hit the acknowledged dead ends that it did in ours. If the space exploration and military budgets of the participant countries are linked (very likely, considering that space has become a much busier place) then that ramps up the available money considerably.
The economics of scale will eventually bring down the cost as well: building a one-off system is more expensive in both the short and long term than starting a production line. And moving launch and production facilities to space as soon as feasible will lower the cost even further, by reducing the cost of raw materials and the need for massive amounts of thrust to leave Earth's gravity well.
As for incentive, a WWII that went much worse might cause the major powers to seek another avenue for their expansion and resource extraction needs besides conquest. Or maybe in this TL everyone had an attack of brains.
4) If we're talking launch sites, Peenemunde is out: it's too far north to be pragmatic, and has too much populated areas around it for a safe launch trajectory. Von Braun's favoured site was Midway Island, in the south Pacific, as it's pretty much on the equator (in OTL it is the Army's Kwajalein Range). Guyana works as an equatorial site too, as does Kenya and India. Basically, you want somewhere close to the equator, with an Ocean either to the south (polar orbit) or east (equatorial orbit).
Simon
Baikonur's at latitude 46 degrees north and that hasn't really hurt the Soviet/Russian space program any. All it means is a need for more thrust and a higher orbital inclination. Peenemunde is at 54 degrees north, so anything launched from there would be at an even more highly inclined orbit. The population density in the surrounding area is not a bonus for range safety so I'd assume the use of Peenemunde presupposes extremely reliable equipment (on the order of commercial airliners today). The author may want to help us out here.