I wouldn't rule out possiblity of shipping nuclear waste to the Moon- but I wouldn't say it's viable at the moment.
Now, I'm not a fan of idea of near term use of space elevators, but if in the future this become viable, possibly one of it's main payloads could be radioactive waste. But you don't need space elevators to ship radioactive into space. Chemical rockets can be made reliable and cheap enough to do this.
To lower cost, first you need to designation some area on the Moon in which these waste can be "stored". You pick an area of say 1/4 million sq kilometers [500 by 500 km area- something about the size of Oregon] and in middle of these area you have 100 km target area which payload crash into at around 2.5 km/sec.
The LCROSS NASA mission is planned launching in 2008 is going to do something similar- impact the Moon at around 2.5 km- they are looking for water in lunar poles:
http://www.aviationnow.com/avnow/news/channel_awst_story.jsp?id=news/aw041706p2.xml
According to article the probe's booster [2,000-kg] will create be targeted to hit a 3 km target area and will create a crater "roughly a third of the size of a football field, and 16 ft. deep," and will create a plume 30-35 mi. above the surface.

So with radioactive material as long plume didn't get much more then say 100 km it's mostly going stay within the 500 by 500 km area. And after billions or trillion of tonnes of radioactive material is "vaporized" in this area perhaps it might even be viable to mine the area [sort of scrap metal mine it]- in any case it wouldn't be hazardous to humans or other lifeforms [unless humans go to the area and live in it].