Nexus Mods Special Edition Cross Over
Skyrim from 2011 (Oldrim) and Skyrim Special Edition (SE) should be seen as two different games. Save games are not compatible and by default, neither are mods, though some mods will be in some cases (with possible warnings though). Luckily, a lot of mod creators have ported their mods to the SE version of the game.
Nexus Mod Manager also won't understand that you want to 'reuse' the mods for SE.
You will have to restart playing and also have to restart installing all the mods you want for the SE. Some mods won't be available nor compatible.
I have both versions of Skyrim and I'm currently playing SE with 250+ mods. About 65% of the mods I used for Oldrim had a SE version that I could use, the rest was handpicked and configured together.