So far this method has worked:
in Settings find Game Centre>Log in as requested (sounds strange I realise)
In my situation I was asked to agree to Apple's T's & C's - and I was offered a Disagree Button - Disagree to the terms
on your next Game play, when the pop up happens, fill in your details; after a period of time I was returned to the fill in your details. My guess is, having declined the T's & C's Apple won't let me sign in. Cancel out of the Pop Up.
Close your game... Re-open the game and Pop-up's are gone
There is a lag in the game where I guess the game is doing a background check on my ID, but the pop-up does not load
Had a fright this morning, as GC said "Welcome Back Me", but think a default load as no Internet Connection. With internet Connection, still no pop-ups
I wonder, if filling in your normal Apple ID, but with a bad password, then keep pressing submit until your account locks has a similar effect?
Good Luck!