Okay…time for a Radiation 101 lesson. In short, the answer is yes, but not badly. Any nuclear bomb testing will release certain radioactive isotopes into the atmosphere. Radioactive Iodines, for example, are frequently seen following nuclear reactions. Cesium-137 and/or Cesium-134 can be by-products of nuclear fission as well.
The daughter product (those isotopes produced by nuclear fission) curve for most US nuclear plants show a high production of Xenons and Kryptons. But US nuclear plants use enriched U-235 which is a thermal fuel. Nuclear bombs generally use either U-235 or Plutonium-239. If Plutonium is used, the daughter products will be slightly different.
Another consideration is where the bombs are detonated. If they are detonated underground, the amount of contaminants they throw into the air are limited. If the bomb is detonated at ground level or in the air (over the Pacific Ocean, for instance), then all the by-products are likewise blasted into the air. Air currents can carry these isotopes quite a ways. When Chernobyl melted down, it released a huge cloud of radioisotopes into the air. The impact of those radioisotopes (how far they traveled) could be seen all over the globe. There isn’t anywhere in this country that you cannot find Cs-137 that is from bomb testing back in the 50’s. Bananas are a great source of potassium, but K-40 shows up in all bananas as well as in people that eat them. K-40 can be found naturally, but only in low levels. Most comes from bomb testing back in the 50’s as well.
But in the long run, Mother Nature has a way of dealing with most of the trash we throw at her. Most radioactive isotopes are relatively short lived. There are the exceptions like I have mentioned…Cs-134, Cs-137, K-40, and tritium (H-3). Another factor to consider is how these radioisotopes might be an impact to humans. Are they getting into some food chain? Are they hazards if inhaled? Do they have strong gamma radiation decays? There are many, many, variables that would make me question whether I should be concerned and what precautions I should take.
Do I want humans testing nuclear weapons? Hell no! Nuclear weapons are one of those things I wish we could go back and un-invent. Am I concerned about the physical impact of one or two nuclear tests? Not really.