Not to worry *yet*.

The plan is to move all of it to Nevada eventually. Wikipedia has what looks like a good summary of the situation with Yucca Mountain.

The UK's plan is put all theirs a few hundred feet underground in Caithness. An area which was a mile deep in glaciers during the last Ice Age, some which dug valleys 3000 feet deep. The next Ice Age can't be more than 5000 years away, maybe only 1000, at which time the waste will not be much less dangerous than it is now.