Knotty1985: Well folks I had this exact same issue, couldn't update without using the web installer (and even that was difficult) and then the client showing it was "offline"
I figured it out and it turned out to be down to DNS servers, normally I use cloudflare DNS servers (for my PC) and OpenDNS for my router.
I tested with my VPN in use and it connected fine so that determined the cause straight away.
If you set them to google's DNS servers it should resolve the issue 8.8.8.8 and 8.8.4.4
Not an ideal solution but its up to GOG support to speak to DNS providers to ensure their app connects over any public DNS.
Screenshot attached shows connections from GOG client without VPN (using cloudflare to resolve) and then on the right with the VPN using another DNS provider to resolve.
Further update: I found my IPv6 DNS servers were the actual root cause, I had cloudflare addresses here as well but they had been corrupted somehow and the last section was wrong, I changed them via the network adapters properties and put the right ones in, and after a reboot GOG now connects properly.
I have google DNS servers as default so I don't think that solves the issue.
I also tried to see if it would work with using my VPN, it did not. The update autofailed at 4.1%.