To make sure things are being installed, just click on the 'Data Files' button and you should see them starting to propagate as they download. So if your mods are not installing, just launch the game and let it sit at the launcher for a few minutes even if it looks like it's not doing anything. Turns out that even though I never got the message of it telling me that it was installing the mods it was still downloading them, I just had no way of knowing. I went online and everything I could find said that when you started the launcher you would see it telling you in the bottom right corner that it was installing and that you would get a message saying when it was done.

I had subscribed to a good 40+ mods in the workshop but whenever I would start the game the mods would never install. Just thought I would share some advice for an annoying bug with Skyrim and the Steam Workshop not working as intended, as I think it's a common problem but I can not for the life of me find people talking about it.

Ck2 steam workshop not working