I can log successfully on GOG web site with my GOG account but when trying to sign up on the CDProjekt forum with it, it prompt me for login endlessly.

Authentication server overwhelmed ?

Same here.
Glad I'm not the only one having this problem.
I searched nearly everywhere since monday but coudln't find anything at all.

The problem occurs on FireFox (endless login) and IE (Login Window not even showing up)

Same for me as well.

Same here, did you manage to fix those issues?

You have to use Chrome. I know it sounds odd, that is the only browser it works with for an unknown reason.

But the reason here seems to be that by default settings mostly chrome allows third party cookies (or you tend on other browsers to disable them quite soon, or something else is blocking). I tried several browsers and even some live OS before getting to that point.
What you can do: on FF set Third-Party Cookies to "From Visited" (if not already set and for sure generally keep cookies for session or let it ask), log in at gog.com and afterwards log in at cdprojectred.com. Or use a Add-on for Third Party Cookie Management (f.e. cookie-monster).
Chrome/Chromium doesn't have the same ability, you'll like use an Add-on or generally allow them. I didn't test yet which would be the best to also control third party cookies, you might want to allow everything and use umatrix f.e.
Internet Explorer seems to have something additionally - so you would need to disable safe browsing.
Generally where to set these options f.e. take a look at http://www.howtogeek.com/63721/how-to-block-all-cookies-except-for-sites-you-use/ (but keep in mind that they advice to block third party cookies totally, yet you want to at least white-list cdpr to gog)