Some of us had a problem like this a long time ago. In a thread here on the forums, someone suggested that you open and turn on a program called Ultrasurf before you open up the ESO launcher, then keep it running while you're using the launcher. It's been a couple years since I used it, but here's a link to the site.

https://ultrasurf.us

It should still be totally free to download and use, and I didn't encounter any problems during or after using it. Frankly, I don't know why it fixed the problem for us back then, but it might still work for you.

Let me know if it works or if you have any questions. I think I actually still have it installed on my laptop so I could help if necessary.

*edit* I found the thread that I originally got this info from, here you go

https://forums.elderscrollsonline.com/en/discussion/81466/launcher-stuck-at-loading/p1

Just scroll down on the first page and someone mentions the ultrasurf solution.