Although the US military was never quite detached from the American space programme, of course the bulk of it was and is handled by a civilian agency, NASA. In many ways, though, this is a historical anomaly, and can perhaps be traced due to the embarrassing failures of several early launches under the auspices of the armed forces, such as the Vanguard programme and the Able space probes (the early Pioneers).
What if these hadn't been so unsuccessful? A POD for this might be that the Soviets' Sputnik fails and they don't get a satellite into orbit for another few months or a year, thus preventing the panicked rush of the Americans to try and match it, and leading to further refinement of those projects.
What would the space race be like if the American end of it had been dominated by the armed forces?