Note that it's not the NuGet packages that are strongly named, but some of the assemblies that they contain. NuGet is just a way to easily get those assemblies into projects, but it does not fundamentally affect the nature of those assemblies.
So the question really becomes: how to deal with version conflicts in strongly named assemblies, independently of NuGet. Normally, binding redirects are the way to go. But if Silverlight doesn't support that, then I'm not sure what can be done there.
There were some early talks of automatically rewriting assemblies with a different strong name (or none at all) to avoid conflicts, but that's fairly tricky business and we never went in that direction.
Maybe others that are more familiar with Silverlight can comment further?