This is my two cents:
The beginning of the timeline is in the Population Growing town, a decade or two before the events of Population Growing, some villagers grow up in that town (source: In Wild World some villagers will say that they grew up in a town with "the coolest monkey" working at a train station.)
Then come the events of Population Growing, pretty straight forward, Gulliver is a sea-farer.
Then, Tortimer moves to the Wild World town to become Mayor there, the shop-owners, who are all good friends with the Mayor, follow him to that town.
Gulliver gets a spaceship and starts flying about space, the events of Wild World occur.
After Wild World, due to serious accusations of the `Boondox Scam` (in which Tortimer gained lots of money from the villagers), Tortimer decided to move once again to a town near `The City`, again the shop-owners go with him.
Also, I should point out at this point that I think the characters you play as in each game are entirely different characters, not the same kid.
Not much changes in the Let's Go to the City/City Folk town.
Eventually Tortimer gets a longing for home and decides to move to a town very close to the Population Growing town, this is the New Leaf town, again the shop-owners all follow him.
Sick of being shot down in his spacecraft, Gulliver returns to sea-faring.
Tortimer remains the mayor of the New Leaf town for a few years before retiring and allowing the player to be the next mayor.
In my mind Happy Home Designer & amiibo Festival are non-canon, but this is all my interpretation, perhaps this is too linear, I think the thing with Gulliver returning to sea-faring after flying a space craft could be wrong, perhaps the timeline goes:
PG -> NL -> WW -> LGttc/CF or something else.
I have seen some pretty wild theories such as it being about a world on the brink of war, and that Wild World had a `holiday-ban` because the town didn't want to get any attention during the war.
I've heard someone theorize that the Wild World & Let's Go to the City/City Folk towns are really camps during the war, considering the Town Gate and such being heavy security, and that New Leaf is a prequel set before the war, whilst Population Growing is the last in the timeline, set in a war-torn landscape.
That is very interesting but of course does have some conflict with the fact that Wild World is referenced in New Leaf, meaning New Leaf can't occur before Wild World but hey, sorry for the long post.
Would love to hear more of your theories!