Firstly, remember that Peter didn't come from the timeline where season 5 takes place, he comes from the season 1-3 timeline. In the season 4-5 timeline, he dies from his disease but reappeared in the season 4 timeline. (See almost all the season 4 for the how and why.)
The events of the season 5 finale created a third timeline that originates from later in the second timeline. Despite the fact that December tells September that "If your plan is successful, we will cease to exist." The Observers still exist, but they are different from the ones we know and more like Michael (Anomaly XB-6783746). This difference doesn't forbid the existence of the original twelve science team that travelled back in time to observe primitive humanity. So it didn't change the events that occurred in the first timeline, like the recovery/kidnapping of Peter and the crossing of Peter from the first time-line to the second one (and the third one).
Walter hinted that the point where the timelines move apart is the invasion of the Observers.
"Because, Peter, the boy and I, we will become a paradox. Nature abhors a paradox. It has to heal itself. It does so by deleting me and the boy at the moment of the invasion. The boy and I will disappear after 2015." - Walter Bishop, explaining to Peter why he must go in the series finale.
The plan seems to only have prevented the Observers from invading our era without altering the previous history.