When we read about the Dementor attack down by the lake in Prisoner of Azkaban, we experience the situation twice. The use of the time turner affects both as both Harry's are present in both situations, we just see it from two different perspectives, first being before the use of the time turner when he believes he sees his father but is actually himself after using of the time turner.
What I don't understand is - the first time the Dementor attack ever happened there could not have been the version of Harry after the use of the time turner to cast the patronus in the first place. The problem lies it the fact that in order to get to the situation where Harry is able to use the time turner he requires there to be a version of himself which has already used the time turner - so if the Harry that used the time turner hadn't saved them, he wouldn't have been able to get into that position.
Presumably the Dementors would have succeeded in kissing him the first time so he wouldn't have been in a state to be able to use the time turner. Even though that is a presumption of what would have happened, the original part still doesn't make sense.