How do I script it (in a two player game where both players alternate using the mouse for one click at a time) so that after the first player clicks, the second player can't click until he or she presses a button on the keyboard, and then after he clicks, the first player has to press another button on the keyboard.

How can I do this without resorting to loading separate levels each time a player clicks? I would prefer them to see the screen at all times.

I imagine you could do something similar to this (in a C# MonoBehaviour-derived class):

bool clicked = false;
bool pressedButton = false;

void Update()
{
   if (Input.GetMouseButtonUp(0))
      clicked = true;

   if (clicked && Input.GetKeyDown(GetPlayerButtonName()))
      pressedButton = true;

   if (clicked && pressedButton)
   {
      clicked = false;
      pressedButton = false;
      SwitchPlayer();
   }
}

`GetPlayerButtonName()` would return the name of the keyboard button that the other player is required to press after the current player clicks.

Hope this helps.