Skyrim has folders in two locations. Your first step is going to be to tell steam to delete Skyrim's files, so right-click it in your library and select "Delete local content...". You'll want to clean up any mod residue by deleting the Skyrim folder at \steam\steamapps\common\skyrim. Your saved games and configuration files are stored in \documents\my games\skyrim. You can delete that folder and the game will be completely uninstalled, although Rabbit51's suggestion to run a registry cleaner is a good idea.
NMM is a slightly different kettle of fish. It's installed where you installed it and it has a cache of the mods you downloaded that may or may not be in the same location as NMM, depending on your installation choices. The preferred method would be to use NMM to delete its cached files. Just tell it to delete each mod individually until they're all gone. This means that its internal mod list will be clear when you start downloading new ones and NMM itself remains intact.
LOOT, TES5Edit and the like are installed where ever you told them to install. I store Skyrim utilities in the Skyrim folder, so they would have gone away with the first step. But if they are in a different location and if you're planning to redownload Skyrim, about the only thing you'll want to do is to remove TES5Edit's backups and leave everything else alone.
I can't speak to Wrye since I don't use it.