Ultimately, make your rules and stick by them. When you re-read your story, read it like a smart fan looking to nitpick. Find any hole you can think of, and then close it.
That's the beauty of being the author--you can decide how to fix your problems. It's very hard to do a time travel story without making mistakes. But usually most mistakes are not unfixable.
Here's the problem with multiverse time travel.
I go back in time, I change something. When I do, I become part of a new universe and the universe I left goes along.
If that's true, the stakes are not very high and the only person affected by a change in timeline is me.
In your story, can I get back to my original universe? Either way, there seems to be no purpose to time travel. In your time war, if my enemy travels back in time, I'm not even going to care because he's not affecting MY universe. He'll just disappear and screw up some other universe.
That's the problem with what Abrams tried to do in Star Trek--it ends the stakes.
The other problem is that Star Trek's time travel rules were completely different from what Abrams tried to do, which obviously causes debate as to what actually happened.
If you follow the Roddenberry rules of Star Trek, then the prime universe was erased. If you follow Abrams' rules, then none of the time travel stories before ST09 have any meaning at all.
Borg travel back in time to assimilate Earth? That's their problem. Sure, Picard and crew would love to save them for humanitarian reasons and because the Borg suck, but THEIR world is fine, based on Abrams' rules.
So you need stakes. Star Trek did a lot of things right pre Abrams with time travel.
I would say yes, one timeline, but come up with a way that time travelers are immune to time travel changes.
Maybe the universe doesn't allow a time traveler to prevent his own existence and while killing your grandfather would erase you from records.