It maybe the English spelling but the institution is real

@LenHazell53 I stand corrected on the "kirtland" thing, sorry, but, question, since even Sputnick was not around, how exactly was this bomb to get to the moon?

@AnneWimsey
I suggest you look in the history of operation paperclip and the long term plans those scientists were roped in to in exchange for pardons for war crimes, that stretched decades ahead.
There was no indication in the OP that the plan was immediately planned to be instituted, and as I pointed out below by 1958 bothe the USA and the USSR were exploding atomic bombs in space at about the same level the ISS orbits now.
Given these facts I have no problem giving credence to the idea that a planned atomic test on the moon is an historical possibility.

In answer to you final question the German V2 rocket could already reach between German and the UK in 1944, Rockets could reach the outer atmosphere by 1958 a year before this report was written sputnik had been place in orbit and Explorer 1 had orbited the earth for the USA and Telstar would be in orbit in less than 4 years.
an unmanned ship would reach the moon in 1959 The first human-made object to touch the Moon was the Soviet Union's Luna 2, on 13 September 1959 only three months after this report was written.