Nuke as in Destroy?
Catastrophic tidal waves, meteor showers of apocalyptic proportions, cataclysmic climate shifts, no more steady calendar, earth would spin off orbit and into the sun, dark nights.
Lobbing an old 1950's weapon of relatively low yield at the moon? Not much, a glass coated hole, and a circular smoothy spot, and radiation...like the moon doesn't have any of that. It would be a visible detonation, if done in the dark side of the waxing or waning moon, but the crater would not be visible or necessarily distinguishable from any other on the moon. Why? No air, A bombs are augmented by the forces of displaced and super heated air. If you don't have that, you have a seismic concussion, a flash of light and head, and short lived shock wave. The plume though, that would be something to behold. Its not at all impossible for debris to reach escape velocity.
Rockets weren't reliable enough though, not wortht he risk of it filing and falling back to earth.
Carl Segan, one of the great minds of our time, R.I.P.