In alt-space timelines, realistic or ASB, I'm always angling for as much international cooperation, specifically US/Soviet of course, as soon as possible.
So it saddens me to say, I don't think this would have worked. If there were something to draw us more strongly into space--if say telescopes or early Lunar space probes showed an alien artifact there, or if Venus or Mars showed signs of being habitable--then I'd be advocating for early US-Soviet cooperation, indeed for this to be formally under the aegis of the UN. (Prior to the PRC being seated as the proper government of China, any time the two superpowers could agree on something, it would be a lock in the UN Security Council, US doubtless being able to persuade UK, France and Chiang Kai-Shek of ROC.)
But without that, the main engine that propelled the US to the moon and the Soviets to develop their sustained orbital program was the mutual rivalry of the Space Race.
A cooperative venture, though endorsed and indeed ordered at the highest levels of both governments, would not have the same compelling urgency to the next levels of power on down. NASA would not get the same massive infusion of funds, nor would the Soviet economic fiefdoms see the priority of diverting scarce resources. With each government pledged to cooperate with the other, either has the option of slowing down the pace, at least within a certain range. It would not do for either to dig in their heels so much and so abruptly as to be accused of trying to pull out completely, but if one side wanted to ask for the deadlines to get pushed back a bit because of technical difficulties, the other would have plenty of factions who would be quite pleased to draw things out further and use funds thus freed up in this budget cycle for some more immediately pressing piece of pork. There's always tomorrow, it's not like someone else will beat us there, will be the mentality.
The upshot would be a much smaller and slower effort on both sides, and the smaller NASA or the various Soviet design bureaux would be, the easier to shunt them aside piecemeal. And meanwhile in other spheres, the US and Soviets will be on collision courses--Vietnam for instance. With the space lobbys of each side shrunken, the likelihood that these other confrontations might lead to scrapping the symbolism of cooperation is much worsened. Having actually gone to the UN for a resolution and a formal international administration might make simply tearing up the treaty awkward, but again the effort can be left neglected, to rot on the vine.
Again, if there were something definite and positive to be gained by getting to the Moon or beyond, the dynamics would be different. A more intense competitive space race for the prize would be in the cards, but also an international agreement to share the prize, trading off the big payoff of getting it all for one's own country against the disaster of the other guys getting there first. Under those circumstances I think both countries would sustain efforts that at least approach our OTL levels of funding; the benefits of cooperation would make up any shortfall and we'd jointly get there sooner and stay longer and do more.
Short of unearthing TMA-1 or the like, or there being a colonizable planet in our solar sysem, I fear the way we did it OTL was the fast track.