Ok, yeah it is kind of obvious since some fans have splurged GP Legend into the main timeline, but hear me out:
So, much like the chronology, it starts the hyper futuristic race we know as F-Zero (200 years after the fatal F-MAX) that took place in 2560. Then, there was the unfortunate event known as the Horrific Accident, aka the Grand Finale, in which Deathborn caused the lethal crash at the last course of the King League, Fire Field.
The Grand Prix was suspended for seven years until, thanks to the persuasion from Super Arrow, the Federation Congress decided to resume the sport with the F-Zero X competition in 2567.
Unlike the main chronology, there was a pilot that seemed to have returned from the dead, Tinsel Steelus. Aside from introducing the Duelists, Yugi and Joey, to the upcoming event (F-Zero GX) several years later, probably in 2573. It was thanks to Falcon that spread her wings and started to race again since the original Grand Prix 13 years prior and that incident in White Land.
Now, GP Legend would be in an alternate continuity, as would Climax, but both the GBA game (see F-Zero: GP Legend (video game)) and the anime (see F-Zero GP Legend) were released at the same time in Japan in 2003, and then the US a year later. So, it's only fair to include both in the same timeframe. Here's the lowdown: In my OC's timeline, they both take place one year after the events of GX. In 2574, the scenarios of the anime went first, but only the first two Laps when Rick was awakened from the 150-year cold sleep and was just starting out; however, Lucy wasn't included in the game's Story Mode, so I theorized that the game must've taken place before the end of the second Lap prior to Lucy and Tinsel joining the Task Force. The anime resumed from there, and of course there were also the events from "Ending it All," (see User blog: TheIkranRider/My_F-Zero_Fanfics) which made some of the circumstances altered in the Japanese/English dubs. They just skewed into a different tangent because of the unexpected cancellation of the English dubs.
And finally, the latest Grand Prix, F-Zero Maximum Velocity, which occurs in 2585 (25 years after the original; I think having it a quarter century after GX was a mistake as it'll take us back to the original). The future's yet to come involving the all-new pilots, machines, and courses, though it's unknown as to how the competition had reduced from the intense, advanced rules to that from the original competition. Perhaps the Federation were having a restart. The future is hard to see.
Feel free to make your own timeline for the F-Zero series, and let's hope it lives on.