Did the Disruption of Sacred Timeline happen during the events of Endgame or after it?

The Sacred Timeline was maintained through and beyond Avengers: Endgame

Agent Mobius is able to show Loki the events of Avengers: Infinity War (including Loki's death) in the Time Theatre in Loki – S01E01. These events are later than the point of history that Loki was seized from by the Time Variant Authority.

Earlier, in the courtroom, the Judge and Loki discuss the 'Time Heist' where the Infinity Gems were returned and the Judge asserts that this has all been incorporated into the Sacred Timeline.

LOKI: I came into possession of the Tesseract because they [The Avengers] travelled through time, no doubt in a last-ditch effort to stave off my ascent to God King.

JUDGE: That’s quite an accusation.

LOKI: Oh, believe me, you can smell the cologne of two Tony Starks. You speak of Time Criminals? It’s they you should be after. Perhaps you could provide me with a taskforce and resources, and I could return and eliminate them for you.

JUDGE: We’re not here to talk about the Avengers.

LOKI: Oh, no?

JUDGE: No. What they did was supposed to happen. You escaping was not.

Loki's new timeline was reset by the Time Variance Authority

When Loki seized the Tesseract from the Avengers and used it to escape, he created a new timeline and became a variant. However he was shortly captured by the Time Variance Authority, who removed Loki and then reset the timeline. Thereafter the variant Loki is seen at the Time Variance Authority (thus outside all timelines) or on missions with them (helping maintain the 'Sacred Timeline').

Sylvie's new timeline was also reset by the Time Variance Authority

The timeline created by Sylvie was likewise reset after the Time Variance Authority brought the child Sylvie in. Thereafter Sylvie shelters on 'The Sacred Timeline' by staying near apocalypses.

The Multiversal Branching causes the 'The Sacred Timeline' to become one timeline among many

When Loki and Sylvie confront 'He Who Remains' in Loki – S01E06, he explains that they have reached the end of the work-in-progress that is the timeline he has engineered all along. He is relinquishing control.

HE WHO REMAINS: We just crossed… the threshold. (CHUCKLES)

(THUNDER RUMBLING)

HE WHO REMAINS: Oh. So, I fibbed. I fibbed earlier when I said I know how everything’s going to go. I… I know… I knew… (CHUCKLES) …everything up to a certain point, and that point was about… seven, eight, nine, ten seconds ago. But now I have no idea. No idea how the rest of this is going to go. I’m being candid.

When Loki returns to the Time Variance Authority following Sylvie's killing of 'He Who Remains', he finds Agent Mobius commenting on the sudden uncontrolled branching of the timeline.

AGENT MOBIUS: That’s, what, 63 new branches in this unit alone? Does he [He Who Remains] want us to just let them all branch?

Can different Timelines Interact?

Loki Director Kate Herron has given some insight into this.

"So, there’s the branches, right, which is like the alternative reality. But then something, you’ll see it. It’s very subtle, but in the very last shot where you see the multiverse, there’s like basically other bigger physical timeline branches. So, it’s almost like these different separate trees that are now connecting."

It’s almost like a bridge. If you imagine the branch, it is like another reality. But if the branch extends beyond a certain point, it will then connect to other physical timelines. […] that last shot we did, there are other like thicker [branches] that are meant to be like our timeline. And there are other timelines like that and the branches are the connectors basically."

The Direct speculates as follows:

The multiverse is obviously something at the very heart of Marvel's Phase 4, and probably long after as well. One of those many branches is likely the one that will cause some problems for Spider-Man in the upcoming Spider-Man: No Way Home. And any number of branches will surely play a part in the madness that Doctor Strange's next film hints at.

Madness is also the best word to describe trying to understand the concept of timelines and alternate realities. It's a tough nut to crack, but there's no doubt Marvel President Kevin Feige and co. are hard at work making sure everything makes sense and is utilized in a meaningful way.

Fans will be taking a deep dive into the multiverse fairly soon, as Marvel's What If...? is set to premiere on Disney+ on August 11. The show will explore multiple alternate timelines where certain events in the MCU went much differently . The What If...? trailer even teases that some of those said timelines may interconnect at some point —much like the bridge that Kate Herron referenced.

So we are being told that there will be interactions between different timelines in Spiderman: Far From Home and Doctor Strange in the Multiverse of Madness. We don't know if the events of The Sacred Timeline up to the end of the Infinity War will be affected or changed by this, however.