von Braun's team lost five years of development time to the Russians between the end of the Second World War and the start of the Korean one. The US military didn't really know what to do with the German rocket scientists and had them stationed at remote Fort Bliss, TX as "Prisoners of Peace" as some of them put it at the time (the Germans needed military authorization to leave the base). They had poor equipment and tiny budgets since building rockets wasn't a major US postwar priority.

America rapidly demobilized after VJ Day and spent five years losing ground to the Soviets in all kinds of technical areas (atomic weaponry is another).

The Soviets had the N1 rocket capable of manned moon missions; the problem was it never worked (four catastrophic failures in four attempted launches). The Soviet rocket and space program never recovered following the death of Sergei Korolev in 1966, and the N1's existence wasn't known widely outside Russia until after the Cold War.

You can see footage of an N1 exploding on the pad at https://www.youtube.com/watch?v=m79UO4HOQmc

Could ask a followup question?

James Michener's novel Space says that the American space program was forbidden to even try to put a rocket in Earth orbit until after Sputnik. I don't remember if it was money, or some existential reason about possible space travel.

In the early years of the Cold War, the US invested very heavily in long-term bombers as their nuclear delivery vehicles, not ballistic missiles. They did, from 1953 onward, invest in several long-range ballistic missile projects, but the Soviets had started intense work in this field from at least 1949 onward.

Why focus on bombers? The US had developed long-range, heavy-bomber technology in World War II (i.e. the B-29) to great success, and the people who ran the Air Force were bomber people. The US also had good geopolitical reasons for favoring bombers — it was a "known" technology and the US had enough allies near Soviet borders that it could relatively easily create a complete "ring" around the USSR. This is a map I made for teaching showing US overseas nuclear bomber and bomb deployments (just bombers, not missiles), from 1950-1958.

The Soviets, as you can see from the map, had the opposite problem — they had no allies close-enough to US soil where they could deploy bombers. Additionally, the US was spending huge amounts of money on bomber defense (radar systems, fighter-interceptor systems, massive computer networks to coordinate responses). While the Soviets did have some heavy-bomber development, the sheer length of the over-the-pole distance to the continental USA, and the difficulties of avoiding defenses, meant that they primarily thought about bombers as something you would deal with only for very close-in targets (i.e. Europe). Instead they saw very early on that ballistic missiles would be the primary technology that would allow them to threaten to continental USA.

By the time of Sputnik, the US already had at least 6 different ballistic missile research projects in relatively advanced stages. They were "behind," but only by a few months. After Sputnik the US quickly moved to rush deployment of missiles (arguably too quickly — some of the early models, we now know, would have barely worked in the event of actual crisis).

As for the issue of the German scientists — by the time the US put the Explorer satellite into space, it was clearly well-known that both Sputnik and Explorer were enabled by German rocket designers. In 1958, a joke was making the rounds: "Said the Explorer to the Sputnik when finally they were alone together, 'And now that we're alone out here, let's speak our own language. Wie gehts?'" Von Braun was, of course, publicly associated with the US program at that point. The Army publicly revealed that von Braun and 117 other German rocket scientists were working on rockets for the US in late 1946. (e.g. "Nazi V-2 Bomb Inventor on Army's Rocket Staff," Los Angeles Times 4 December 1946, p. 8)