I have run NMM (and Steam which doesn't want to update either) as an administrator but that doesn't make any difference. I could play Skyrim on the PC w/o the NMM but, I don't know, some habits die hard.
I am having the same issue and I think it started with version .4413 At the same time Steam stopped working so I couldn't access Skyrim (if that makes any sense) I gave up on Steam and downloaded a nosteam version of Skyrim (I do own a legitimate copy). But when I upgraded to 4413, now I can't login... so no Nexus Mod Manager.