As far as the Soviets were concerned, the early space race was 100% about displaying a technological superiority to the West, which equated to a moral and philosophical superiority, a vitally important tenet of Communism. They really couldn't have cared less about the science.
The Americans, while still driven largely by competition with the Soviets, at least were somewhat interested in the scientific gains to be had. Von Braun's Army funded Jupiter/Redstone missile was a capable and fairly well developed vehicle in 1956 and was fully capable of launching a satellite at that time. Unfortunately he had been ordered to ballast the vehicle with sand in order to keep it from eclipsing the cobbled together official "civilian" Vanguard program. So, an American satellite in 1956 using a Von Braun Jupiter was a very realistic scenario.
At no time was there ever a "missle gap". Technologically, the Americans were always ahead and only stupid political interference prevented us from being first. The Russkies got ahead only by taking enormous risks, especially with the manned program, and just got plain lucky. When Kennedy took office, the political interference largely ended and the engineers were left to work their mojo. By the last flights of Mercury we were back in the lead and by Gemini were were surging rapidly forward. The Russkies never got beyond the politics and Khrushchev's constant interference distracted their program into a distant 2nd place.
With all this in mind, I see America being first with Explorer 1 as going one of two ways:
1. The Russkies take their lumps initially, but plod on in heightened secrecy, with their eye on a bigger prize. They perfect the R-7 launcher and launch a series of Sputniks. They initiate the Vostok program at about the same time. The Americans, meanwhile, kind of coast along believing that they are ahead of the Soviets. Mercury is started up pretty much as per OTL, but with a lack of urgency. An accelerated Vostok program puts Yuri Gargarin into orbit in January, 1961 with two more flights by summer. The Americans are caught completely flat footed and stumble along with Mercury, not managing a sub-orbital flight until December, 1961. Scott Carpenter finally orbits a year later, John Glenn's mission having failed to reach orbit (he aborts but survives). The American overall technological superiority reasserts itself at this point and the situation stabilizes and curiously reverses itself. By 1964, the space race is virtually indistinguishable from the OTL.
2. The Soviets goad themselves into greatly accelerating their program due to the need to upstage the Americans. They suffer numerous failures and in April, 1961 manage to kill Yuri Gargarin in an ill advised orbital attempt. They virtually shut down the program at this point and divert the funds to the military. They do manage to secretly orbit several crews later in the deep black military Almaz/Soyuz program, but deny there was ever a space race. The lack of the perceived competition from the Soviets cause the American program to proceed at a leisurely pace. The Mercury flights are completed and the Gemini program finishes in 1966 after six missions. The program converts over to unmanned missions and probes are flown to the Moon, Mars, and Venus by 1970. No manned lunar program is started or contemplated.
So, scenario 1 results in big changes initially, with the OTL reasserting itself eventually. Scenario 2 has the Soviets pretty much backing out of the space race altogether and the American program fading away due to a lack of competition.
Dave