You don’t need a three stage Apollo rocket that can lift 150t to LEO for your ICBMs.

That area of launch ability was functionally complete in 1960 with Redstone and possibly a bit of showing off in the guidance phase with Gemini.

ICBM development started decades before moon landing, same in USSR. It’s true that first space rockets were reused ICBMs, but the tech diverged after that. ICBMs use different fuel because they need to be stored for years with full fuel tanks, for quick deployment. There is no such requirement for space rockets. Space rockets are also much more powerful - you could fit dozens or even hundreds of nukes on Saturn V which is impractical - you don’t put all your eggs in a single basket. Saturn V was never intended to be used as an ICBM.