In my case I just left the issue and it was resolved by itself. My initial suspicion was that it was caused by the EGS network issue, though checking it now there doesn't seem to be any issues.
Looks like this issue was related to one of Epic's servers that weren't responding. Since I posted this I went through the logs for the Epic Launcher and found that a specific URL was returning HTTP 503 responses. It seems to be working perfectly fine now. I'm not sure if I'm allowed to post URL's here, but the specific one in question was the "friends-public-service-prod06" URL found on the "Troubleshooting" screen of the Epic Games Launcher.
I downloaded this from Epic Games Store on Sunday (free game) and have been playing since. I also played earlier this morning, no problem. When I try to play now, I keep getting the message "Requires purchase of the base game". I don't know if this is Epic's fault, I've restarted my machine and also checked the EGS support scope and they state to log a request with the developer of the game itself. Any ideas?