Since
Eddie Thawne killed himself
there would be no
Eobard Thawne coming from future to kill Barry's mother.
So the time line will be altered drastically: remember Eobard said to Wells that he only built the particle accelerator in 2020. So how is it even possible for the Flash to exist?
When Eobard Thawne ceased to exist, it should have reset the timeline so that Barry's mother was never killed, Barry grew up with his parents, and Harrison Wells' particle accelerator didn't come online until 2020 (resulting in the creation of The Flash).
However, that didn't happen. Instead a paradox ensued, the result of which created a singularity above Central City. Barry and Firestorm were able to close the singularity, but it seems to have created rifts into at least one alternate reality, Earth-2.
Harrison Wells warned Barry that when you go back in time and change things, the universe has a way of working itself out in unexpected ways. Eddie killing himself would have been a complete rewrite of a lot of history. So instead, the universe decided it could solve all that nonsense by creating a black hole instead.
Despite the supposed never existence of Eobard Thawne, Barry and his Super Friends retain full knowledge of his life and actions.
By being at the locus of the temporal event, the Flash Team can remember Eddie but the universe at large may have erased him from the timestream because of the reshuffling being done at the time. Wells, in his effort to change things created a Predestination Paradox. The Flash would always exist. All he did was make him exist EARLIER in his career. This sets off an entirely different chain of events from what Thawne knew from his future. This is why you don't play with time travel kids. You could erase yourself and everyone you know and no one would be the wiser!
Different writers handle this paradox in different ways. Some subscribe to the popular 'Many Worlds' theory which implies, connections are made to alternate, parallel universes and as such, the possibility of the new parameters have worked themselves out SOMEWHERE in the multiverse. Thus the conditions made by whatever foolishness that has taken place here CAN exist and will settle out over the timestream, eventually.
Other writers change the times stream the traveler is on and thus when he returns to his future, the events which have taken place in the past have altered the nature of his future such that he is divorced completely from it. He may even cease to exist. Looper tries to do this with varying levels of success.
There are more ways of attempting to adjust for paradox in science fiction and most of them are unsatisfying to say the least. If you don't mind the headache, I will direct you to a short list and why they do or don't work.
Classic Time Travel Paradoxes and How to Avoid Them: It's not perfect but it tries to explain why time travel and writers should never be allowed in the same room.
