What happens in the ongoing timeline when Cage dies?

Everything should be reset. The power resides on one of the aliens, and when it senses it is loosing one of the captains, it turns back time. It's not Cage's power, it's the alien. This alien would be alive in both timelines, and he'd know he pulled back time, it wouldn't make sense.

So, I'm guessing it's all in a single timeline.

There is no evidence for or against a Many Worlds Interpretation of time Theory in the film. We can assume, if the MWI is valid, that the majority of what we saw results in doomed time lines, but with MWI, there is an infinite number of possible saved time lines as there are doomed. We just don't see.

if instead we take it to be like BTTF, each time travel event, results in a unique timeline, in which only one timeline where humanity is saved (along with n timeline where the mimics died and rebooted). It would be significantly less than infinity.

The last choice is like XMEN Days of Future Past. Information is sent back, and the timeline that was changed no longer exists.

Thematically, it has to be the last one. Because the story puts through no alternative. It strives to show that time is not fixed, that you can change the past and thus the present and the future. Had they went with MWI, all of Cage ' s trials would be meaningless. Nothing could be changed Rita and humanity would always die, to that Cage.