As the title says, what happens if Vanguard 1 launches before Sputnik 1? For the sake of argument, suppose that this is because Soviet leadership develops similar fears to Eisenhower re: the legal regime for satellite overflights, and so slow-walks their satellite program so that the Americans will launch first and establish that free flight is permitted (more of a concern for the Soviets, in fact, because their satellites must cross American territory, whereas American satellites do not). The Naval Research Lab, as IOTL, doesn't know what the Soviets are or are not doing, so they go ahead with their program more or less as they did, and end up launching some time in 1958, followed by Sputnik 3 (which is Sputnik 1 here). What next?