Only Wound rolls and only when it succeeds on a 5- but rolling a natural 6 would change the Save or Injury roll.There is the same issue with any skill/ability/trait/tactic card that allows you to reroll a hit, wound or injury roll.
Problems can only arise if the player is given information too soon and that information can influence a choice they have to make. You will always reroll a failed Hit or Wound roll so knowing beforehand what was rolled for Save and Injury doesn't change anything. And since there is nothing after the Injury roll (that is fast-rolled), you are not given any extra information that could change your choice.
Horizontal fast-rolling creates more of the same issue btw, as you know all the result of all the Wound rolls at the same time.