The guy just uploaded turn order mechanism.
I haven't seen someone translate it, so here goes. These are the 10 rules outlined.
1) Action time is set by the Pokemon's speed stat according to the table. Turn order is decided on at the start of battle, action speed works as a wait time of sorts, when it hits 0 the turn comes up.
2) If wait time becomes 0 at the same time, the Pokemon with a higher Speed stat will go first. If the Speed stats are tied, then the player side will go first.
Afterward, priority is given to the Pokemon that did not just act.
3) When ambushing, the wait time on the player side will be 0, and the wait time on the enemy side will be 1. After escaping, the wait time on the enemy side will be 0, and wait time on the player side will be 1, if you are to fight the same opponent again. (TL note: I'm unfamiliar with Legends' mechanics and haven't played it so I'm not sure if "surprise attacks" are things the player can do or have happen to them)
4) Moves using Agile Style and moves that target the player side can decrease the wait time on the player side, moves that target the opponent side can increase the wait time on the opponent side
5) Consecutively using Agile Style attacks will reduce the effect.
6) Power Style moves uniformly have an effect of increasing wait time by +4.
7) Paralysis has an identical effect to reducing action time by 1.
8) Action time is +/- 0 for recovery items, and add -3 wait time for buff items.
9) When switching, if there is a difference in action time, the enemy side will act immediately. Otherwise, the enemy wait time is set to (enemy wait time - action time). (TL note: I'm not sure if this is an accurate translation)
10) After a death, the Pokemon sent out inherits the elapsed wait time.
Following that is a detailed explanation in the rules.
Rule 1 is easy enough to understand with the visuals accompanying it, I think. In the example given, Crobat has an action time of 7 while Petili has an action time of 12. Crobat acts at 7, 14, 21, 28, etc, while Petilil acts at 12, 24, 36, etc. Consequently, the turn order goes Crobat (7) - Petilil (12) - Crobat (14) - Crobat (21) - Petilil (24) - Crobat (28) - Crobat (35) - Petilil (36).
Rule 2 is also fairly easy enough to understand. In the example given, the opponent Crobat and player H-Lilligant have the same action time. If Crobat has 250 Speed and H-Lilligant has 245 Speed, Crobat will act first. But if H-Lilligant also has 250 Speed, it will act first instead, because it's the player side. However, this can be dangerous if the opponent uses an Agile Style attack and the player does not, as their wait time will be lowered enough to get two consecutive actions. In the video example showing this, the Golbat and Mothim had identical speeds, so the player's Golbat went first, but the Mothim's Air Slash allowed it to act twice in a row, leaving the player no response.
Rule 3's explanation shows that if you initiate an ambush (is that right?), no matter the difference in Speed you will get the first turn. However, it's still possible for the opponent to get consecutive turns even with the same action time. In the example given, Crobat and H-Lilligant have the same action time, but if Crobat uses two Agile Style attacks, then it goes H-Lilligant (0), Crobat (1), H-Lilligant (7), Crobat (7), Crobat (13). The opposite is true if you flee an opponent and refight them, so you could use this to your advantage against Alpha Pokemon.
I got kinda burned out translating at this point, but for the final rule 10 regarding KOs, the example involves a Mothim with an action time of 10. In the first example, Roselia has an action time of 10, and Petilil has an action time of 11. In this case, after Roselia is KO'd by Mothim, Petilil inherits the action time of 10, and gets to act at 21. Mothim gets to act at 20, so it goes first again. However, if Roselia's action time is 9 instead of 10, then Petilil inherits the 9 and acts at 20, getting a turn that it otherwise would not get.
There's a lot more to it than I translated but I think people can generally get the gist of it. Well, it's something like that anyway, I'm not certain I grasped the explanation precisely.
I also found a list of moves by action time, but I'm not sure it's fully accurate. Here it is anyway.
+4: Quick Attack, Mach Punch, Bullet Punch, Ice Shard, Shadow Sneak, Aqua Jet, Wave Crash, Baby-Doll Eyes
+3: Extrasensory, Fire Fang, Thunder Fang, Ice Fang, Twister, Zen Headbutt, Bite, Rock Slide, Astonish, Air Slash, Iron Head, Icicle Crash, Icy Wind, Bulldoze
+2: Swords Dance, Nasty Plot, Double Hit, Focus Energy, Shelter, Acid Armor, Iron Defense
+1: Bulk Up, Calm Mind, Victory Dance
+/- 0: Everything else
-1: Phantom Force, Mountain Gale
-2: Hyper Beam, Giga Impact, Roar of Time, Dark Void, Chloroblast
But I'm not sure about this because I thought Extreme Speed has improved action time.