Gravitational potential energy of moon = 7.35 * 10^29 J Energy of largest nuclear bomb (50 megatons) = 2.1 * 10^17 J

So the bomb is over 10^12 times weaker than the moon, affecting its orbit and the tides by one part in a trillion. Quite unnoticeable. Source: Wolfram Alpha

Either that, or it would blast the moon out of Earth orbit and send it on a journey across the galaxy.

Source: Space 1999

I wonder if a more interesting effect would have been the spewing of debris into earth orbit. The moon doesn't have an atmosphere to damp the ejection of material from an explosion.

Although, I suppose this also happens if a meteor hits the moon.

Its a reasonable question. If you did it such that the explosion was on the earth facing side, you would impart a significant amount of energy to the material ejected. Some of that would move in a retrograde motion to the moon, thus reducing its orbital velocity and lowering its orbit relative to the moon. Other material would have the opposite effect, and material ejected perpindicular to the orbital plane would find itself in a very strange orbit indeed (probably reconnect with the moon later as the orbits eventually intersected).

I can't imagine though that you'd get enough of it into an orbit that intersected the Earth's atmosphere to do any sort of damage. You might have some great meteor showers as it entered the atmosphere but probably no chunks big enough to reach the ground.

Nothing, and no. https://www.quora.com/What-would-happen-if-every-nuclear-pow...