Is it possible to go from a Backgammon opening position to a Nackgammon opening position in 6 moves? And if not, how many turns would it take?
I don't have a lower bound on number of turns. However, 6 is not enough. (I'm assuming you meant 6 total turns, not 6 turns each.)
You can work backward just from the second player finishing their cleanup. Assuming WLOG that the second player is red.
Red's final move both have to be to 23 or 24 from Bar or 24, and can't take any white pieces. White's final moves both have to be to 1 or 2 from Bar or 1. So, in its first two moves, red has to take a minimum of two white pieces in the white home board while moving a piece from 13 to 2 and a piece from 6 into a takeable position. If done as part of the second move, that takeable position has to be 1 or 2.
Every combination I can find that allows red to do this in two moves leaves white unable to fulfill its similar conditions on turn 2.