The nuclear testing in the south Pacific happened in the 20th century and picard crashed the enterprise in the 24th century. The radiation would have subsided enough at this point to be negatible at best.Well if they can travel faster than light, shoot energy beams, use matter antimatter reactors I would wager that some nuclear radiation is not going to be all that hard to defend against Not to mention the obvious fact that radiation disperses over timeWell Gravette Island doesn't exist but if it did it would serve the purpose of letting the crew go unnoticed and "stay out of history's way". Protecting the timeline was more important that the crews safety.
I don't know if you've ever been to the south pacific, but the nuclear testing was confined to a few small islands (namely Bikini Atoll) thus the effects of the fallout were largely isolated to those islands, no indication was given that Gravette was one of them. Not to mention, the events of First Contact occur nearly a hundred years after the last of those tests, the radiation's effects would be largely non-existant by then. In fact some of these islands are already well on their way to being habitable again today.
The fallout from Castle Bravo was larger than the U.S. expected, because yield was larger due to the fact that they did not know precisely how the effect of the lithium-6 and lithium-7 isotopes fusion-boosted the yield of the device. Radioactive ash rained down on many square miles of the surrounding ocean, and then was carried by the currents, and on a Japanese fishing trawler, the Daigo Fukuryū Maru, which was located outside of the U.S.-declared exclusion zone, poisoning and killing members of its crew ProfessorTrek 00:24, July 31, 2011 (UTC)
The Enterprise DID crash in the 24th century, the Enterprise D crashed on Veridian 3 in Generations. The Enterprise E never crashed into anything except the Scimitar in Nemesis. Picard ordered the crew to the island in 2065, not the 24th century, then had the escape pods recovered somehow towards the end of the movie.