I don't think it will and I don't think it should.

And you already named the reason for that. Most people wouldn't upload their mods to several places anyway. If they wanted to do so they already can (Nexusmods, etc.) but most people simply don't do it. And that won't change with a Galaxy Workshop.

I doubt it. Maybe they could integrate mod sites like nexus or mod db into galaxy 2.0 and make the process easy to use, but i dont seem them creating a closed up system.

No I don't and no they shouldn't. We're already at that place where Steam has fragmented the formerly platform-neutral modding community via the parasitical Steam Workshop under the guise of "convenience", and fragmenting it even more is not helping. I think what they are doing now (linking to Nexus in posts like "The best mods for x game" like this is absolutely fine. Maybe they could even expand on that, have a whole sub-forum dedicated to mods and mod guides with links direct to Nexus or ModDB both on forums and direct links from within Galaxy. It sure beats trying to "Out-Steam Steam".

(disclaimer: I know little about Workshop)
I think they should get something if it's useful to remain competitive, however ideally it wouldn't be "closed" like Steam's apparently is. You should be able to download mods as you please (unless that'd put a big strain on GOG's servers). I'm not sure they'd really want to rely on a secondary hosting site, in case the site goes down. I'd say a lot depends on how much Workshop inclusion sways people to Steam over GOG.