President Eisenhower was keenly interested in getting a satellite up for a very specific purpose--he wanted orbital surveillance of the Soviets. He was alarmed at the risks involved in aerial surveillance missions but also unwilling to go without the information that overflights brought him, thus the desire for orbiting surveillance satellites.
But, he also figured that if the first spacecraft the USA launched were a spysat, the Soviets would assert a principle of airspace sovereignty extending out to infinity, claiming the right to shoot down any satellite passing over Soviet territory at any altitude whatsoever.
Hence, Vanguard, as a nominally civil and scientific program. The plan was, Vanguard would launch first, and the Soviets would say nothing against the obviously harmless and completely public program. Then after a few of these successes, other American rockets could start launching other satellites with a minimum of fanfare or publicity. The Russians might start crying foul at that point, but having established the principle of limited airspace and the international character of orbital space, Eisenhower hoped, their objections would have limited force, and eventually they'd reflect that they too could have free surveillance of the USA from satellites of their own. Since Eisenhower offered the Russians the "Open Skies" proposal whereby each nation would be allowed to send verifiably unarmed scout planes over the other freely, I think he'd anticipate and accept reciprocal Soviet spysats as inevitable, fair, and not undesirable, since honest information about what both sides were doing would tend to cool down war fevers.
In the event, the Russians beat Vanguard into orbit, which was not what Ike wanted, but served the purpose of preempting Soviet objections to Corona spysats much much better. Any Soviet-launched satellite would surely pass over US territory (whereas US launched craft might completely avoid passing over Soviet territory, the US being at a much lower latitude) so since Khrushchev did not ask US permission nor even give a warning, the Soviet themselves were by implication asserting the free passage of orbital space as a principle; all the USA had to do was agree.
Once Sputnik was up the restraints were taken off von Braun's Huntsville team and, quietly, the Corona program was cleared for launches.
Early Corona attempts were a long series of failures by the way, I believe it took them ten launch attempts to finally get a spysat up that took useful pictures and then returned them recoverably to Earth. But I believe that first successfully recovered set of pictures was delivered to Eisenhower before he left office so he could see the fruits of his policies. However much money it cost to put up that finally successful rocket, and factoring in the costs of the failures that had preceeded it, not one American risked being shot down over Russia to get these pictures; with the legal metaphysics of "who owns orbital space?" settled in favor of an analog with the open seas rather than extended national sovereignty, by the Soviets themselves, there was much less risk of an international incident leading to war than sending over an American spy plane.
It's just wrong then to say that Eisenhower had no interest in space; for this specific purpose his interest was strong. It was just that, with the best minds he could find working on a secret rocket project, it was necessary to mask this with a second, open, project, and he did allow that project to bog down because too much White House interference in what was supposed to be an ivory tower scientific program might raise questions as to the true nature of Vanguard and just why the President cared so much.
On the other hand, I am not sure just when Eisenhower was convinced that a series of spysats would be a possible and good solution to his intelligence dilemmas, at a guess I'd say no earlier than 1954. I'm also not sure just how early the idea would have percolated upward in the bureaucratic chain of command to land on the desk of any Presidential intelligence or science advisor, nor even how early space enthusiasts would have made the proposal for that mission seriously. In the 1940s, probably there would have been little confidence that an automated machine could have done the job and such notions would have been linked to the idea of putting a human being into orbit, which of course greatly complicates the matter by requiring a method of landing that person, along with their negatives, safely. The mass of a capsule that can carry and sustain a person is greater. (Though actually spysats, especially early ones, needed a lot of mass too, for good telescopic camera lenses, and as technology progressed allowing a spysat of a given capability to be smaller, the ambitions increased at a faster pace and the trend was for spysats to get bigger and bigger in fact). So that would have made early surveillance satellite proposals that much more blue-sky and remote--and meanwhile, since we are putting at least one human into space, mixed the surveillance mission in with many other objectives. After all the first human launched into orbit would be big big news, lots of luck trying to keep it secret!
So I don't think it was just cultural conservatism that marginalized the space program until the latter half of the 50s. It was realism that pointed out that a satellite, unless it was just a dumb hunk of metal just to score prestige points (or a passive comsat like Echo) would be a manned mission, much harder to achieve.
Could a crash program determined for some unspecified reason to accomplish that as quick as possible have done so before Yuri Gagarin actually became the first man in orbit in 1961? Or even before Sputnik in 1957? I suppose, with enough effort, it might have been done. I do think we'd want more advanced rocket engines than the alcohol-burning V-2 engines! With lots of them clustered, with lots of stages, with an escape system for a capsule for the pilot because launch failure would be quite likely, maybe this might have been done by 1950, using kerosene-like rocket fuel and liquid oxygen, or possibly the "storable" hypergolic room-temperature acids used by the Titan rockets that launched Gemini OTL--these latter are very nasty stuff though, I personally would push for ker-lox.
So, maybe a decade more advanced, but the capabilities of a given system would be less than their OTL counterparts a decade later, and their costs much higher.
And one still has to think of a motive for a government to put that high a priority on it. Especially when the decision-makers involved would presume that gradual progress on all fronts, particularly linked to the development of ballistic missiles that both Americans and Russians were keen enough on doing, would solve many problems for them and eventually bring the capability within their grasp without a crash program, so the hypothetical alt-motivation would have to be of an emergency character that could not wait patiently like that.
The best I can think of is, an alien spaceship strands itself in orbit and the mission is to get to it and claim it first.
