The bug

This may be a parity issue  but relates to MC-249875. In that bug, Allays could have a delayed reaction before deciding to do certain actions, such as throwing items or following the player.
This bug was fixed in 22w42a, but Allays still sometimes have very noticeable delays before following the player. Compared to Bedrock, where Allays follow the player as soon as they're out of range instantly every, time.

Even a few seconds can be crucial for a mob that already has a hard time keeping with players, so it's important they track bonded players as soon as possible.

Comparison

Although not consistent, this clip is a good example of how they can delay following for a few seconds:
AllayDelayedFollowing120.mp4

This clip also showcases at the end an example where the Allay decided to not follow for several seconds, and only ended up "wandering" close to me: AllayDelayedFollowing120_2.mp4
And finally, here's a clip from Bedrock edition to showcase how instantaneous the mob is able to switch to follow mode as soon as I'm out of range:
AllayFollowingBedrock.mp4

How to reproduce

1. Spawn and Allay and give it an item
2. Move out of range multiple times. Sometimes it will instantly start following you, but sometimes it will take a few seconds for it to notice
3. Attempt this at various speeds and ranges to see how it effects its ability to keep up