Should you return to the Source after meeting Feo Ul, Tataru's dialogue
does change to indicate she's met Feo Ul and that you should go back to the First and let her worry about the Source. The quest in question, "Travelers of Norvrandt" actually changes a multitude of small talks! The Scions have kept many of their allies in the loop regarding the missing Scions, it seems.
Tataru, Matoya, Lambertient and Arenvald all change after this; Lamberteint doesn't quite believe the Scions are in another world, Arenvald worries for Alphinaud, and Matoya asks us to tell Shtola to come visit when she returns.(I do wonder why they've been updating Lambertient since 4.4...) Of those four people, three of them have the same small talk after completing 5.0's MSQ—Tataru's dialogue, however changes. For clarity's sake, here are Tataru's 3 states:

Originally Posted by
Tataru, beginning of Shadowbringers
Forename, y-you're back! Have you already finished saving the world!?
...Not quite? Well, if you need to get back to the First, it should be a simple matter of touching the beacon. It's at the same place we found it!

Originally Posted by
Tataru, after meeting Feo Ul
Forename! I was going about my business the other day when a feisty little pixie suddenly appeared and gave me a report about your doings!
It certainly sounds like you have your work cut out for you. So just leave things here to me and focus on getting things done in the First!

Originally Posted by
Tataru, after defeating Emet-Selch
Not much to report on our end, Forename. The Eorzean Alliance is still locked in a stalemate with the Empire.
As ever, in the Archons' absence, I'm keeping the ship that is the Scions of the Seventh Dawn sailing smoothly along. I run a tight ship, I do!
Admittedly, I learned this after I'd personally completed Shadowbringers by way of poking around GarlandTools and the like—but I do have a vague memory of checking in on the Rising Stones before Amaurot to see what, if any, diaogue changed and was amused/happy to see at least Tataru's did while the other Scions in the Stones didn't.
Another somewhat fun fact, is that the only other time dialogue on the Source changes is after you slay your first Lightwarden, Temulun (Cirina's grandmother) changes her small talk to the following:

Originally Posted by
Temulun
The light within you shines as brightly as before. More brightly, even. But your journey is far from finished. I shall pray for you.
Everyone else on the Source, other than those I've mentioned here, just seem to have a flag for whether or not you've completed relevant quests (the lv80 job quests) or Shadowbringers (the more mundane Scions). The citizens of the First seem to have a multitude of states, typically dealing with whether you've restored night to the area they reside in, if the Light's come back, and if it's Lakeland or the Crystarium there's even more nuance... it's honestly surprising and I'm ashamed to say I missed most of the interim steps!