They also thought about using nuclear bombs as propulsion for rockets in the space program, called Project Orion.
Thank God that never went anywhere.
The fission detonations would only begin once Orion was above the atmosphere.
What killed it was a failure of political will. Otherwise, we would have been to Mars by 1965 and Venus by ‘73. An interstellar version was also planned with a crew of 265 or so.
We would have had successful colonies on both the Moon and Mars by now.
Instead we settled for third or fourth best alternative and ended up with the dangerous antiques: the Apollo and Space Shuttle. Both now dead.
As for blowing up the Moon - now or in the 50s - not a chance.
The largest fusion bomb ever detonated was by the Russians yielding 90 plus megatons. Many craters on the Moon were created by detonations in the millions of megatons for comparison. And the Moon is still there ...