Let's deal with the fundamental issue.
At the end of the day, having weaponized satellites they don't control orbiting above them is something neither superpower had the slightest willingness to accept. If anything weaponized is flying, they'll treat every object as a potential risk, and the need to avoid crises will outweigh any desire for weather satellites and alternatives to cable.
That's not to say that militarized space has to kill space development, though. It only kills certain kinds of space development.
The initial Soviet line that airspace extends beyond the atmosphere had to be modified in order to send Sputnik into orbit. But they didn't have to flight it over the United States, per se.
Let's say that perception of current crises or tensions differ in the USSR immediately before Sputnik's launch. Rather than risk inflaming the situation, the satellite is shot so that it fails to cross American soil. They could either do this by setting it on a semi-synchronous orbit - where it would cross over India, the Middle East, Africa, and the Indian Ocean. I'm not sure if the specs existed yet for that, but failing that option, they could also choose an orbit that would allow it to circle the globe several times before passing above an American state. A built-in mechanism could initiate reentry before that point.
That's not a hugely likely turn of events, but it seems quite plausible. If they repeated that policy until the Americans got into space, the US would probably have followed suit. Neither side would be likely to break the pattern and so obviously take a threatening act against the other. The great thing about this POD is that it compounds - the longer objects are in space avoiding the "airspace" (spacespace?) of the powers, the more military-ish hardware will go up, and the more violating the practice would appear overtly hostile.
At the same time, because neither power could strike directly at the other from objects orbiting semi-synchronously, there just wouldn't be the same impetus to demilitarize space. It's possible, given the right politicians at the opening of the period, that treaties merely formalize understandings that the two countries will not over-fly each other, rather than mandating what can fly. Since initially they're the only two countries in space, it's a conceivable arrangement.
US launches from Florida and Soviet launches from Kazakhstan would essentially carve out two massive orbital spheres of influence. The US would have the ability to deploy its weapons across the Southeast Pacific and most of North and South America, excepting the extreme east of the latter. The Soviet satellites, by contrast, would overlook the entire basin of the Indian Ocean and Middle East. Given difficulty with launch sites and the geopolitical situation, the skies above Europe and West/Central Africa would likely be empty.
Obviously no nation is going to be terribly comfortable with weapon platforms orbiting over it, but the superpowers aren't going to stop going into space, so compromises will be made. Alliances like NATO will include agreements permitting orbits. Smaller nations will be appeased to curry favor or ignored by the superpower "above" them. The more influential or useful states may negotiate or demand for the powers to take paths around them - China would likely end up a good example of this.
The Commonwealth, French, as well as China and India, are likely to view a presence in space in much the same way they viewed possession of nuclear arms in OTL - you don't really count unless you're up there. As a result space development will be at once more limited/localized and better funded/supported.
In general I see the consensus developing that if a state has rights to all the space above it, but only once it has the ability to get there itself.
Byproducts.
Expect a stronger Commonwealth, certainly. The Australians for one practically share a border with Russia in this scenario. The Canadians will be not so much concerned as annoyed, as well. Former British colonies from Malaysia and Singapore to South Africa will be severely uncomfortable (or enthusiastic, if still under white rule). India...?
India is the only potential space-flying nation that will unavoidably be criss-crossed by another. They might turn to the Commonwealth to deal with this, trying to get into space themselves. Given economic limitations and the perennial issue of Pakistan, it'd be a long time before they got properly in orbit. In the interim I'm inclined to predict a parallel to our Non-Aligned Movement, but with a much stronger base. Absolutely one of its tenets would be the right of any state to declare its airspace sacrosanct. Given physical constants that's impossible. Still, the constant challenge and perceived threat would foster cooperation and a greater sense of shared interests of otherwise totally disparate nations. I'd expect it to accomplish more on the world stage, even if it is limited to tools like boycotts and mass walk-outs from the UN floor.
This avenue would get horribly messy in three or four decades, but not particularly in a way that'd threaten nuclear war.
Right now I'm imagining a college student in the '70s railing about Western cultural imperialism in preserving European spacial territorial integrity, but not that of the impoverished Third World. He has a pony tail. Awesome.
I love it. Alright. This needs to be played with. Clearly.
On it.