Currently, Saved Instances shows which characters you've completed a Random Timewalking Dungeon with, but this resets daily.

When you're running dungeons throughout the week on multiple characters, It would be extremely useful to know which characters have completed the 500 badge turnin.

You can do the quest once per event, per character and each of them have their own quest ID.

• BC - The Swirling Vial: https://www.wowhead.com/quest=40168
• WotLK - The Unstable Prism: https://www.wowhead.com/quest=40173
• Cata - The Smoldering Ember: (H)https://www.wowhead.com/quest=40786 (A)https://www.wowhead.com/quest=40787
• MoP - The Shrouded Coin: https://www.wowhead.com/quest=45563
• WoD - The Shimmering Crystal: (H)https://www.wowhead.com/quest=55499 (A)https://www.wowhead.com/quest=55498
• Legion - Whispering Felflame Crystal: https://www.wowhead.com/quest=64710

Right now, the only way to know if you've completed these for the week is by logging into each character and running the completion check script, so being able to check this at a glace would be amazing.

I hope you consider this addition :)
Thank you.

I've completed a MoP Timewalking dungon on every character shown here, throughout the week. I'm not seeing any reference to it on my logged-in character, or any of the others o.O Am I looking in the wrong place?

They're tracked under weekly quest. You can hover on it to show details.

Could you try /dump SavedInstances[1].activeHolidays and reply prints?

I'm on US servers. Timewalking just ended, and the weekly reset also just hit, so I might have to wait until the next one and get back to you.

Edit:
Will come back on next Timewalking - WoD: 3rd Jan 2023

/dump SavedInstances[1].activeHolidays

No Completed Dungeons:

[1]={
  [1971]=true,
  ["Random Timewalking Dungeon (Warlords of Draenor)"]=true
}

Edit 1:

Completing a dungeon without turning in the badges doesn't show anything in the weekly quests.

Edit 2:

Turning in the 500 badge quest still shows nothing in my weekly quests for SI. This is a mistake. It does show up

I've completed the quest The Shimmering Crystal and it's correctly showed under Weekly Quest.

Oh you're right, it does say it on the same character. But try switching to another character and checking your alts, because that's what the original issue seemed to show in a previous comment's screenshot.

You can see that it doesn't show when your alts have completed it, only the currently logged in one.

Quest is still showed after switching to another character. Could you use that macro after you switched?