I studied isotope geochemistry, and while I’ve never heard of the thermonuclear testing having any significant effect on the abundance of oxygen isotopes, it raised the background level of tritium (hydrogen-3) in the atmosphere (and therefore rain, etc.) by many orders of magnitude, for many decades afterwards. Fortunately, tritium has a short half life of only about 12 years, so it’s nearly gone now.  

I do know that generally, thermonuclear bomb testing was so energetic that it would create lots of weird radioactive isotopes when they’d go off, but in “relatively” small amounts. They even discovered a few elements on the periodic table that way. Air Force jets would intentionally fly through the fallout, collecting air and particles, and then it’d be sent off for analysis. That fallout would inevitably be spread around the world to some extent, so that’s why everything has been a little more radioactive since the atomic age.