It’s not that they came here centuries after the nuclear tests, because even if they have FTL that is the shortest time frame that makes sense unless a nuclear explosion causes some signals that some how move faster than light which both violates physics as we know it and would be really unlikely to go unnoticed based on just how much data we have from nuclear blasts and nuclear reactors.

And in general the gamma burst form a nuclear explosion is so minuscule on galactic scales that it’s unlikely they can even pick it up over the background noise.

A spectral analysis of the atmosphere of the earth and the detection of fission borne radioactive isotopes is a much more likely method which means they would need to be actively observing the earth to detect it and still the same light speed capped timeline is in effect.

All of this is. You can either live in something resembling the real world, or LARP your life away in a weird fantasy. If I were going to do the latter, I wouldn't pick this particular fantasy, it's boring and unoriginal. Try reading some good science fiction instead.