An earlier POD...
I tend to think that the real problem was NASA, and the whole approach to human spaceflight in general. There was a wealth of activity in the 1950s regarding space exploration, with a variety of approaches under examination. Eisenhower, however, was deeply committed to a civilian space program and saw very little value in military involvement beyond recon and spy sats (the Corona project). This, combined with the extremely poor political skills of the Redstone Arsenal team (the Germans), led to numerous promising ideas being discarded early on, and a 'doubling down' of bets on losing systems (Vanguard anyone?) with strong political backing. Ironically it was the duplicated effort, and multiple solutions to a single problem that were most needed at the time, as the necessary technologies had not (and probably still have not) yet matured.
Let us try something different then...Eisenhower dies/retires/loses interest/etc in 1955 or so, and the various services (and their many contractors, who were really driving things before about 1957) continue to compete to put satellites in orbit and launch manned spacecraft. There is very little doubt that some form of Man In Space Soonest would have put a human in space by 1960 (maybe a year earlier, but that is questionable), and if each service was free to pursue their own (perhaps multiple?) approach, we might have seen several different architectures in place. Clearly the Air Force wanted to continue their rocket planes (and the Dynasoar/MOL) for instance, and the emphasis on surveillence systems would have led the Navy and Army to continue their emphasis on ballistic launches. Various contractors at the same time had their own programs under examination, and while they might not have borne fruit initially, McDonnell Douglas, Convair, Boeing, and Northrop all had plans in place for the mid/late 1960s when it was expected that a civilian market would emerge.
Obviously some of these systems would fail, turn out to be dead ends, or just end up wasting resources, by it is hard to imagine a bigger dead end than Apollo, which consumed incredible quantities resources and brought back little more than rocks and (arguably) some technology. Without NASA having the only keys to the manned space bandwagon (i.e. several different space programs - civilian, military, commercial - all continue), it is likely that the differing needs of different customers would result in a considerable infrastructure being left in place. There would probably not have been a moon landing in 1969, but we probably would have a few small space stations (along the lines of MOL, maybe a few others), several different launch systems (Dynasoar was particularly promising, though others were certainly practical) and most of all a launch rate and diversity of booster types that would have given us a sustainable infrastructure to continue development.
OK, to summarize:
1955: Eisenhower (for whatever reason) drops his objections to a more diverse approach to space. The military (all branches), civilian govt, and commercial interests all continue to actively plan for real involvement in space over the next decade
1956-57: First satellite launch by US Army, followed (possibly) by Convair, the USSR, the USN, the USAF...
1960: First manned spaceflight (likely USAF, though US Army was a competitor), followed by several launches (including USSR) over the next year
1964: First long-term space flights, followed quickly by small space stations and associated resupply/cargo launch systems
1968: First permanent space stations, polar orbiters
1973-1974: First lunar expeditions
?
Sorry for an (even for me) unusually wordy post...