Tested related scenarios:

• Looting applies if you kill mobs with a harming potion while holding a looting sword.
• It does not apply if you kill a mob with TNT while holding a looting sword.
• It does not apply if your tamed wolf kills a mob while you hold a looting sword.

Note that projectile looting is reported as a bug in Java Edition at MC-3304.

Thank you for the additional information - the repro steps and video examples we have now should be enough, and the issue is being looked into by the team.

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Looting seems to be applied to the harming tipped arrow, but not the tridentMinecraft-2021-09-22-23-25-43.mp4

This is a severe nerf to Bedrock players, as others have noted.  For example, with drowned-copper farms bedrock players cannot make use of the reinforcement mechanic present in Java to get faster drop rates.  Factoring in the nerf from this bug makes it really hard to get renewable copper for all console and mobile players.  Hopefully it gets a hotfix in days rather than weeks.

Attached a video showing this with looting 255 sword(obtained by inventory editing). Looting effect is clearly not applied to mobs killed by projectiles My Video.mp4

I have found that I can get looting if I shoot a harming tipped arrow into the ground, pick it up, and then kill mobs with it while holding the looting sword. Picking up a tipped arrow in this way seems to reset the entire stack it is added to, so perhaps something with the inventory related to MCPE-70591 is happening?

However, it could be that I'm just getting looting from the harming effect and not from the projectile directly. I'm not sure how the game handles that technically.

@Magdalena here is a hopefully easier explanation.

How to reproduce:

1. Make a superflat world.
2. Gear up with a power V bow and a looting III sword and harming II arrows.  (EDIT: Harming arrows were suggested to ensure 1 hit kills but you should use normal arrows because of GoldenHelmet's comment below) 
3. Place a mob spawner and use a ghast spawn egg to make a ghast spawner.
4. Stand about 10-15 blocks away so you can quickly switch weapons while the arrows are flying. For the next step you must have the sword in hand when the arrow hits. 
5. Fire an arrow at a ghast and quickly switch to the sword before the arrow hits. Do this repeatedly for a couple minutes. Keep a count of the number of ghasts killed. 
6. Go pick up the loot.
7. Optional: for controls, do the same number of kills by hitting the ghasts with the looting III sword, and again using the bow without switching weapons mid flight. Count the drops for each and compute the average number of drops per kill for each. 

Expected result:

The number of drops when switching to the looting III sword after launching the arrow will be the same on average as killing the mobs with the looting III sword directly. (This has been the behavior for as long as I can remember, up to and including version 1.17.10)

Observed result:

The number of drops is the same on average as killing without switching to the looting sword. 

You can do this for any mob, but ghasts give a clear result because the gunpowder and ghast tears show a big difference in drops with or without looting.

Attached the video to repro. Keep in mind that you need to repeat the step on Java to compare the drop rates.
2021-09-22 22-46-39_Trim.mp4

We could not reproduce the issue. 
Could you provide a video of this happening? If the file is too big, upload it elsewhere and link it here. 

Added context. Trident killers using the looting effect are a required part of Minecraft Bedrock edition gameplay to balance the game further towards Java. Java edition has a much larger mob spawn rate, and mob cap, which allows all of it's hostile and passive mob farms to be incredibly faster than Bedrock Edition's farms. It also helps make up for the lack of Bedrock Edition not having sweeping edge, which allows for Java players to quickly kill large numbers of mobs with a single attack, getting the looting effect applied to all mobs killed. Bedrock does not have an equivalent to this.

Trident killers are used to bridge this gap.