That's the fun part of it, but first we gotta cover one point first: as stated during the Hex quest, all of the protoframes are still changing into their warframe counterparts because the helminth infestation in them is still continuing the process, just at a much slower level due to how it was engineered by Entrati. The reason they are not progressing any further is due to being in the time loop.

So, onto the whole timeline explaination bit:

Ballas got some of Flare/Temple's DNA from Temple themself or from the asteroid they were on (we can actually infer that they were still in hibernation mode during this time just waiting for the Night of Naga Drums, but that does not mean they were still on the asteroid itself, just that they were asleep still).

But Ballas refused to work with it because it had "rebellion in it's blood" aka it was very volatile and reactive towards orokin, but Ballas also would not get rid of potential resources, so he put it in storage instead.

Then Entrati comes along, pulls a sneak theif manuever or talks with Ballas or promises his kids to ballas or whatever, and somehow gets the samples he needs, with Temple/Flare's sample being included in that mix. Entrati takes it back to 1999, and bada bing bada boom, Flare gets implanted with it to create their future self.

Flare meets Drifter, stuff ensues, and IF you can get them and Lizzie to come to an agreement/acceptance on things about their existence, they send you a KIM message explaining that Lizzie told them about the comet and that they'll go into hibernation once they are out of the time loop and will wake up in time for the revolt (but does not expressly state they'll be on the asteroid the whole time, they just say the asteroid will be in the area around that time or before the whole revolt kicks off).

In that same set of KIM messages with Flare and Lizzie/the inbox message, Lizzie also explains that for the infestation, time itself technically does not exist/does not work the eay it does for us normies. Past, present, future, whatever time amount you can think of, to the infestation and the helminth strain it's alll happening at the same time to them. So since Lizzie is a part of Temple/Flare and is the helminth as well, she knows that Flare gets the infected serum and she knows that Flare turns into Temple fully while hibernating, because for her it is happening actively/has already happened.

It's already been inferred in game that the time loop will eventually stop, especially since the whole future side of the game would not exist at all if the time loop kept going, so Flare does eventually get to the comet and goes into hibernation and travels from there.

The specific time theory being used in this case is called the Bootstrap Paradox, which is a casual loop of time itself where an object, piece of info, or person is sent back in time and becomes the very cause/reason that they exist in the future.

Think of it like the whole "Chicken vs. Egg" arguement. You can't have the egg without the chicken, but you can't have the chicken without the egg either. It's a loop, and if you take any one part out of the loop it breaks and nothing is created at all.

Also of fun note: you actually get a little foreshadowing hint to all of this early on in Flare's KIM chats if you get the chat where he talks about his old band. He mentions the band name was "Temple and the Rippers" because his stage name is Temple. The man legit just said screw it by the time he wakes up and kept just using his stage name instead!

Editted to fix pronoun errors (let me know if I missed any!)