Radiation is not an issue. Bikini took the brunt of the testing. There's two chains of islands that make up the RMI, Rataak and Ralik. The largest population base is in Ebeye and Kwajalein, both located 250 miles away and in the Ralik chain (same as bikini).
The capitol, Majuro is in the Rataak chain, some distance away.
You can still dive Bikini atoll about once a year. There's a live-aboard that go's thru there. Its quite an expensive and long journey via a live-aboard dive boat to do so.....used to be cheaper and more accessible. Given you can dive there, the waters are relatively safe as the ocean is huge and able to dissipate pollutants more readily than the soils.
What makes Marshall islands rock unique is its lightweight and porous structure. You cannot harvest from the ocean side, as most of the time, the waves are too turbulent and would wash away anything lightweight over time. The lagoon is much much calmer and shallow.......more likely to find rock here.
As for the original question, Kwajalein has a robust ES&H department that conducts testing throughout the year, as well as the Corp of Engineers. Radiation levels are low enough that you can dive 10 days a week, 10 hours a day you'd be unaffected. Some of the bigger problems are fuel oils leaking from the numerous ship wrecks from WW2, with the Prinz Eugen being the biggest offender. I think there was a recent effort to remove some of the fuel oil by mechanical means.
Pics for reference are from ocean side ( I know, i said earlier, no live rock is found on the Ocean side, this is the exception) in an area referred to as the "Japanese tide pools" where portions of the solid reef were removed to add onto the island. This type of reclamation also happened on Majuro to expand the airport runway capabilities.
Fun fact, coralline algae actually grows on the bottom of the live rock, not the top naturally.
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