For anyone that still has an trouble with the timeline, here's a good rundown of it:
The events of S5 directly follow episode 4.19, Letters of Transit, which filled in much of the blanks, and the S5 premier gave further context.
The finale of S4 ended in 2012. Bell had been defeated, but had managed to escape. Olivia was pregnant, and the team had caught themselves a break. However, September appeared to Walter to warn that "they" were coming.
Between 2012 and 2015, life was more or less peaceful. Peter and Olivia raised their daughter Henrietta***, and Walter and September worked on a plan to defeat the Observers, who they knew would invade at some point. After devising the plan, September partitioned the plan out of sequence within Walter's mind as a countermeasure against mind-reading; to re-order these fragmented thoughts, Walter would need a Transilience Thought Unifier.
In 2015, the Invasion began, with Observers from the 2600s appearing en masse, transmigrating from the 27th Century to the 21st to escape their environmentally hostile future Earth. Three year-old Etta was separated from P/O, through means we don't yet fully know. A rift formed between Peter and Olivia; Olivia went to New York to take on the Observers with Walter, but Peter stayed behind, obsessed with finding their lost daughter. At some point, William Bell joined the fray, presumably aiding the Fringe team to some capacity. The Fringe team fought valiantly for the next two months, but the Observer forces overwhelmed mankind. Walter sent Olivia to Grand Central Station to retrieve the Thought Unifier, but she was in danger of being captured, so she Ambered herself. Walter and the gang lost communication with Olivia, and thinking the fight lost for the time being, Peter/Walter/Astrid/Bell Ambered themselves to evade capture as well.
The Observers install themselves as the dominant force. The surviving humans - or Natives - were forced to live under a strict regime. Those who sided with the Observers in the Invasion became Loyalists, and Fringe Division was taken over, given the autonomy to police the Natives, but answering to Observer authorities. And a small Resistance formed, continuing to oppose Observer rule. The Observers ruled for the next twenty-one years, and at some point, Etta realized the old Fringe team's members were her relatives, and joined the Fringe Division.
This brings us to the events of 4.19, which take place in 2036. As was shown, Etta used a black market connection to find Walter. With her Fringe Division superior and Resistance ally Simon Foster, they brought Walter out of the Amber. Seeing as his mind was degraded, they re-connected his stored brain tissue to restore his faculties, and they tracked down and freed Peter and Astrid (at the cost of Simon), while Walter severed Bell's hand. In 5.01, they tracked down Olivia, though Walter was captured, and his interrogation degraded or destroyed the fragmented memories September had placed in his mind, making the Thought Unifier useless. And this is where we are going into tonight's episode (5.02), with the plan lost and the team forced to start from square one.
The reason there is a large "gap" between 4.22 and 5.01 is because the story has been told out of sequence. S5 is a continuation of 4.19, which takes place twenty-four years after 4.22. Much of the events that happen between have to be told anecdotally/in passing, though as we have seen in the opening to 5.01, there have been some flashbacks (in this case, the moment of Invasion in 2015).
The jump in chronology can be a little jarring, but hopefully, the above outline will address your questions.
- Props to Omniscient_Jay again