Locked out of Google accounts by Safari fault.

Going through the rigmarole of trying to sign a replacement Mac Mini into my Gmail and other Google-related accounts. It takes my email address, but when I click Next, it bounces back, as if nothin is there. Any clues? It seems it will only use Safari for this and I don't normally use Safari. Firefox is my chosen browser because I have accounts across Mac, Windows and Linux. Need all 3; don't need Safari because it's monolingual, so it hasn't been used for a while (probably since the last "new Mac" setup.)