The disposal on nuclear (or radioactive) waste depends on the country. In the US, there are compacts or arrangements between groups of states for disposal of low-level nuclear waste. See - http://www.ag.ohio-state.edu/~rer/rerhtml/rer_61.html
The waste, encased in steel drums, is buried in a land-fill. Handford site and Barnwell (South Carolina) are two such sites. See - http://www.downwinders.org/llw_facts.htm
High level waste (HLW, e.g. irradiated pressure vessels) is also buried, encased in some material. See - http://www.oversight.state.id.us/waste/highlevelwaste/
The DOE has a separate facility (WIPP, or Waste Isolation Project) near Carlsbad, New Mexico to deal with HLW from the DOE and weapons program.
Then there is the spent fuel from commercial nuclear reactors, which is currently stored in wet storage (spent fuel pool) or dry storage (casks) in an ISFSI. see http://www.nrc.gov/reading-rm/doc-co...res/br0216/r2/
In Japan - http://www.enecho.meti.go.jp/english.../disposal.html
Plastics would most likely be limited to low level waste. Radiation would not be signifiant, but as for plastics, it would produce some disintegration and perhaps cross-linking making the plastic brittle. Off-hand, I am not aware of any research into the disposition of long-term storage of plastic in LLW or HLW.