I would say when I would have traded queens and at least one minor piece. If you did not trade queens, then activate it when there is only one more piece other than the queen.
When is recommendable "to activate" the king?
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My general rule of thumb is that when the opponent is down to 1 rook and his queen is off, the king needs to be activated. The main exception is this: if there are no open files, or you control all the ones that are open with your own rooks/queen, and you have a space advantage, the king needs to be activated immediately. The king is important for preventing enemy pieces (mainly rooks) from infiltrating behind the pawn formation.
In most endgames, the king is a more effective offensive piece than a bishop or knight. Getting the king to the middle of the action is often the difference between winning and losing.
I'm totally a beginner, so sorry for this silly question. I have always heard it is good to move away the king from the center of the board at the opening, by castling it as soon as possible. On the other hand, i have read it is good to have an active king (near the centre and out of the back rank) at the end of the game.
The question is: how to know the time to activate the King has arrived?