Sorry for double post but this may help other people with this unresponsive Game Center problem.
I contacted Apple Support by phone (amazingly it was quite painless and quick).
The Apple support guy was extremely helpful and knowledgeable.
He suggested the problem in my case was a software conflict and not an issue with iOS 9.2 and so far it seems he was correct.
So I did a full backup onto my pc through iTunes first.
What I then had to do was reset my iPad back to factory settings with no added apps or games.
As a guide;
Connect to iTunes, connect iPad to the pc, hold Home and off switch on iPad together. Keep holding these two buttons (for a good 30 seconds in my case) until you finally see an iTunes icon with an image of the lightning cable appear on your iPad screen. Then follow the prompts in iTunes on the pc and then click "Restore".
In effect you are trying to get your iPad back to its pure original state, and with iOS 9.2.
Once that is done then check if Game Center was working, which in my case it was!
So while I now have the laborious task of reinstalling every game and app I have one by one and checking Game Center each time, it does mean that I will eventually find what app or game causes the issue with Game Center. (Although by using iTunes and having all my previous games and apps backed up means no bandwidth usage for me to do this).
If I do find out what app or game causes Game Center to stop responding I will post my findings here.
Hope this in some way helps others.