I have been harping on this for a long time now.
After months and months of research(by playing the game), I have come to the conclusion that clicking on the middle of the screen when it is grey or a person is spinning, will speed up the game, particularly in 4 player Alternate Shot games, but actually it's in any game that has 2 or more players.
The simple reason for this is that WGT expects a response from your client(aka a click) to let them know you are still in the game. You might be there, but if you don't click on the screen it won't acknowledge that you are there.
I would suggest that the WGT client send a ping(computer talk) after every shot, no matter who is shooting. This would, I think, eliminate grey screens altogether.
I might be wrong on the computing solution, but I am NOT wrong that screen clicking speeds up the game and stops the "spinning" that delays the game.