Hi there!
As the title says, this BUG is happening to Boosted characters(When you bought Dragonflight or The War Within, and received a level boost with it (actual max level -10).
This issue is happening on retail since (or around that time) the 20y Anniversary event started.
The BUG in short is: Boosted characters will NOT receive the quest from killing the last boss of their weekly first timewalking dungeon.
(It is still happening after the anniversary event ended)
–What you can read below is the story of how i figured it out–
How i know it’s related to boosted characters: I only use ONE boosted character, my rogue, and this is the one that started to NOT receive this quest.
I helped my friend to level up a character from Lv1 through dungeons, and this character IS receiving the quest (im not naming the quest, because it is different with each timewalking event)
Then, he wanted to level a character for skinning and told me, he didnt yet use his level boost from buying The War Within, so he used it for his new skinner character. We leveled it right now up to 80 from lv70 with Pandaria Timewalking and once again, a boosted character, did NOT receive this timewalking last boss quest.
So i came to conclusion, it seems only our BOOSTED characters are NOT receiving this quest.
Then he told me, but why my old worgen warrior wont receive the quest either?
and i told him, it might be boosted by buying a previous expansion (most likely Dragonflight), and we verified it is certainly a boosted character aswell, by checking if his character completed the worgen starter zone 's first 2 quests which he should have completed IF he didnt boost the character, and so we saw, he DID NOT complete it, meaning its most likely also boosted.
So this is 2/3 garanted boosted characters (but surely the 3rd is boosted too) which are NOT receiving this quest.
We also sent bug report about this.