Upon configuring Firebase project, downloading google-services.json file and uploading it to Flutterflow, it doesn't get used and FlutterFlow seems to keep using the same keys. (Edit: Regenerate Config step was performed after uploading new JSON config to Flutterflow)
I checked that it's the same key that FlutterFlow uses for any new project - this key creates problems with Google Sign-in. The only way I was able to fix this was to replace the json file in the downloaded code by the one from my Firebase console. However, this means that the problem will persist if the app is published directly from FlutterFlow or FlutterFow directly builds the apk (confirmed).