CD confirms what I thought was correct and that is you can save your shot at anytime until you play your next shot. It does not matter how many shots are played by other players, you can still save your shot until you make another shot.
The dialog boxes for both save and share options have 30 second timers on them though, but this happens whenever you click on the 'save replay of last shot'.
Basically, you have PLENTY of time in which to save your shot so stop panicking, hehe.