It takes at least another turn for Logistics to actually flow over this road. So you still get reaction time.
You sure about that? I'll go back and double check, but as I recall when my opponent made a beautiful strategic maneuver to separate my forces into two disconnected valleys, because I was sitting on a pile of logistic points I was able to sudden-build an impossibly long road from one valley over a high mountain chain several hexes wide and into the second valley. Voila, supply was flowing and troops were eating next turn.
-Mark R.