During WWII the German V2 program did achieve a couple suborbital flights, but these were very basic, and there wasn't really a path from there to orbital flight, the fuel wasn't very good, the engines were bad, and the guidance was basically nonexistent.
Sputnik and Explorer required engine and computer technology that simply didn't exist just a few years earlier. And using similar technology both countries were just barely able to get things to the moon, flying past into deep space or slamming into it with no way to slow down. The ability to do complicated maneuvers required study, but even if we go back in time and gift them a modern textbook and a copy of Kerbal Space Program to practice on these probes were effectively dead stick by the time they got there anyway.
In technological principle the earliest a crewed moon landing probably could have happened is 1965 or 66. 1966 was the year both the first successful orbiting and landing probes reached the moon. The rocketry and computing technology existed, and Gemini had demonstrated life support capable of keeping a crew alive long enough to complete a basic lunar mission. The Saturn V program hadn't yet solved a couple key engine problems but there were Advanced Gemini programs that offered a risky and complicated option for a lunar landing using only Titan launch vehicles and already existing systems.