Scala Tic Tac Toe Game
Whether Scala or any other language, it's not a good sign when you have big blocks made of pasted/copied lines (tttFormat, board and atPlay).  There is some logical structure to those repeated lines and you should code that instead of copying lines.
There is a bug: the middle row is inverted (654 instead of 456).
The line asking for the player move should probably be in the while-loop.  Just my opinion.
To keep good "separation of concerns", the player input should be read in the main loop and passed to the method that updates the board.
The nextTurn followed by nextTurn at the end to get the winner is "hacky".