It is interesting how the consensus is that the USA would be 5 years or more short of an ability to shoot down any satellite, and that if the Nike-Zeus approach is really the best, it takes a megaton nuke to do it!
This suggests to me though more a lack of will to develop ASAT that early, than that it really would be so hard to do it. My belief has been that killing a satellite is much easier than trying to stop something coming in on a suborbital ballistic trajectory (which would be some kind of ballistic missile warhead).
A suborbital warhead is going to be coming in on something that looks like a distorted parabola if you plotted altitude versus range. It would in fact be a segment of an elliptical orbit. This means that as it closes in on one's defended territory, it is going to be coming low but also fast. Hitting it with something light will not change the trajectory much; it is easy for the enemy to harden the missile, especially if it is delivering a nuclear warhead or some such.
A satellite, once launched, is in a fixed orbit, usually circular, which means one can predict pretty exactly when and where its track will cross over one's own airspace. One has hours or if necessary days or weeks to plan the take-out
A satellite killer therefore might be quite light and simple; the idea is to collide with it, and the nature of its trajectory allows for very exact and relatively simple prediction of its location. I am thinking of some sort of buckshot type "warhead;" strew some BBs in its path.
The satellite will be going at a speed of around 7800 m/sec; to be much slower it has to be very high up. If it has an elliptical orbit it could be much faster when at low altitudes. If we can just launch some heavy chaff in its path, the very fact that we don't seek to put the junk in orbit but have it arrive at the right altitude essentially stationary, at the top of its arc, increases the killing velocity. The task of launching a sounding rocket to merely reach a height of several hundred kilometers is much easier than putting the same mass into orbit at that height.
But, none of the OTL systems that might be improvised to shoot down a Sputnik discussed here work that way. All have been extrapolated from prospective ABM weapons which have a much tougher task. As far as I know no one was talking about simple buckshot type satellite killers until Star Wars (SDI) evolved to consider the "Brilliant Pebbles" approach in the 1990s.
Now, as I understand it, Eisenhower would most definitely not want any American forces shooting down any Sputniks. He might just possibly want the capability, but it doesn't look like either his administration nor Kennedy's invested heavily in developing that ability.
The reason Ike would not want us to shoot down any Soviet satellites is that he was counting on the development of US satellites to spy on Soviet territory. He desperately wanted to establish the principle that orbital space is analogous to the high seas, territory of no nation, free for passage for all. Therefore it would be possible to put a US satellite over Soviet land to get a good look at what goes on there, and somehow get that picture back to Washington for analysis. The ability to do this was something he, and I think any rational US President or member of the intelligence community, would value very highly. The scariest thing about the Cold War from the Western point of view was uncertainty about Soviet capabilities. All intelligence available would put the estimate of their capabilities low--but what if the Kremlin masters were cleverly hiding some successful initiative, and could do much more than it appeared? Keeping that kind of secrecy in place in the West would be much harder, given nations where citizens have the right to roam around as they would and take pictures of whatever they please. (The Air Defense Command interceptor base in Florida my father's USAF career centered on had a US highway running between the HQ and other core base function areas and the flight line; a back way to the HQ area had no checkpoint and went right past the radar complex). In the USSR there were no such rights of easy and unquestioned travel; without satellite intelligence it might be entirely possible for the Soviets to develop entire complexes unknown to the Soviet public.
The need was not so symmetrical, since Soviet intelligence could learn a lot without such exotic means. And the Soviets knew they were behind, terribly behind, and feared that if the Western powers knew how weak they were they would preemptively attack. Therefore for the Soviets satellite capability might have seemed optional and a frivolous further cost, whereas denial of it to the Americans might have seemed worth running some risks.
Therefore it was a great gift to Eisenhower's priorities that the Soviets chose to orbit satellites over the USA without ever asking any sort of permission; it meant that if the Soviets were to suddenly take a position that American satellites passing over their land were some sort of violation of their rights, that they would look irrational and selfish and legally in the wrong to the world at large. Tolerating Soviet overflights was no guarantee of safe overflight rights for American satellites, but it was a big step in the right direction.
For a US President to authorize the shooting down of a Soviet satellite, particularly the very first one, even if we had the means to do it, would b pretty self-defeating and stupid, I'd think. It would be far less crazy for the Soviets to establish the precedent of shooting at foreign objects flying over their land, but of course very bad PR for them to do this after first flying their own satellites over the USA.
If American satellites had been first, and then the Soviets were to shoot down the first one that poked over their territory, it would then be less irrational for Americans to threaten to shoot down any Soviet satellites going over US territory (or that of our allies, such as NATO Europe or Japan). Given Americans have lower latitude launch sites, it would be possible for us to launch well south of Soviet "sky space" while the Russians would find it impossible to put anything in orbit without violating American sky space, and thus the USA could shut the Soviets out of space. Of course it would still be possible for the Soviets to shoot at American satellites that did not cross over Soviet space and thus turn orbital space into a hot zone of the Cold War, interdicted by mutual hostility.
This would be a major escalation of the Cold War, making an outbreak of full on war more likely. It would do very little, essentially nothing, to stop both sides from developing ICMB capability to devastate the other side, and in the short run, before 1964, the USA would have the advantage (though this might not be known for sure in the West), able to wreck the USSR while the Soviets could do only partial damage to the USA. (This would still be very bad; we'd surely lose NYC and Washington, and anywhere from half a dozen to two dozen other major US targets, but the USSR would in such an exchange be completely flattened while enough US power would remain to dominate the whole remaining world. Western Europe and Japan/South Korea and possibly Taiwan would be as bad off as Russia probably).
Seeing this, and knowing their own vulnerability, the Soviets would have some motive to seek a detente, as would Western powers, and I suppose negotiations for terms permitting some launch of satellites would follow, but they would be highly restricted. Both sides would invest in heavy batteries of ASAT systems, that could to a limited degree serve as ABMs as well, though poorly. One does not need nukes to kill satellites unless they are hardened to an insane degree while nuclear warheads would be much harder targets.
For either side to start playing the ASAT game would be pretty disastrous, and I think this explains why OTL ASAT seems so hard to do--neither side wanted to be accused of planning to shoot down all the enemy's assets in orbit so any ASAT development tended to be very low key. Both sides benefit more from an "open skies" in orbit regime.
Therefore such an ATL is low probability, and it would have been extremely irrational for the Americans to start the game of shooting down satellites, unless of course they planned to immediately follow through with preemptive war on the Soviet bloc, with all the costs that would entail.
It might be worth following through on developing an ATL where the Soviets shoot first, and thus the Space Race is greatly stunted and postponed. I would guess that in such an ATL, space travel of any kind would be severely limited even if a treaty regime to allow limited orbital assets to both sides is developed, and that to this day no one would yet have landed any astronauts on the Moon.