I answered a similar question on the sister site, but the really short answer is "The Speed Force" makes everything OK. Because it transcends time itself, a lot of "plot holes" in the story can be explained away as Barry being special because he can access the Speed Force. (The same is true of the comics, for what it's worth.)
The longer answer is: there was only ever one Barry Allen. What changed was Barry Allen's life from 2000 to 2016. There have now been four versions of Barry's life history
- In the original original timeline, Barry grew up with a mom and dad, and eventually became the Flash sometime around 2020.
- In the Season 1 timeline, distant-future-Thawne travels back to 2000 and kills Barry's mom. He then impersonates Harrison Well's and makes Barry into The Flash in 2014.
- In the "Flashpoint" timeline, 2016-Barry-Allen travels back to 2000 and stops distant-future-Thawne from killing Nora Allen. He then returns back to 2016. In this timeline, young Barry Allen slept through the whole mess, and grows up with both parents, but somehow still has his Speed Force powers by 2016.
- In the post-"Flashpoint" timeline, 2016-Barry-Allen travels back to 2000 to stop distant-future-Thawne from killing Nora, then travels back a second time to "fix" his mistake. This Barry we know very little about, presumably we'll figure it out as the season unfolds.
Note that 2000 "child" Barry Allen always grows up until 2016 "adult" Barry Allen in every timeline, and he always gets his powers in every timeline. The difference is what his childhood is like during that 16 years.
It's also important to note that, as a Speedster, Barry (and Thawne) have some degree of natural immunity to time paradoxes, because of their access to the Speed Force. This is why, for example, they both remember the previous timeline when no one else does. It also explains how Nora's still alive, if Barry never knew he needed to go back and save her from Thawne, etc.
As far as we are told, young Barry grew up not knowing there was anything odd about his life, right up until he reached the point in the timeline where Season 2 Barry "comes back", and from then forward he has his Season 2 memories. As the episode goes on, it appears as if "new" Barry's memories are starting to replace "previous" Barry's memories. This is the new timeline trying to impose it's will on Barry, again with some vague connection to the Speed Force.