It depends on what you mean by visible and what you mean by space. If you mean low earth orbit, then they would be easily visible. It would appear as a bright flash (even at 200 or so km, it would still be bright enough to be uncomfortable, maybe even dangerous). You would then see a cloud form around the site of the detonation. The size of the mushroom cloud would be a couple Kilometers, so it would be visible, but not easily so.

From somewhere like the moon, the flash would still be visible. A few years back, a small meteor hit the moon with the energy of around 5 tons of TNT, less than 1/3000 the energy of the bomb dropped on Hiroshima. This was visible from the earth without a telescope. It was about as bright as a 4th magnitude star, putting it as visible from somewhere with decently dark skies. This means that it would be very clearly visible. If my back of the envelope math is correct (and I'm remembering the physics correctly) this would be something around magnitude -4.5, which would be pretty clearly visible, if on the night side, similar brightness to a bright planet. I'm not sure how visible it would be considering that the bomb was dropped during the day so it would be against the brightness of the Earth. Finally, a significant portion of the energy of a nuclear bomb is released as x-ray, ultra violet, and gamma rays, which are invisible. The X-rays and Gamma rays are absorbed by the atmosphere, heating it, so they would still contribute to the visible brightness.

In short, if you are close enough to the Earth to be in a stable orbit, you would probably be able to see the explosion. If you're in low orbit, the mushroom cloud would probably be visible.

Also, this question might be better suited to /r/askscience. While it's a historical event, this is hypothetical and uses scientific calculations, rather than historical accounts or evidence.