One idea I have is to say that when the teens finally make it back to Earth, they realize they've slipped into an alternate timeline, and this can be our adaption of Flashpoint. I'm also envisioning a major villain in the form of a time remnant of Barry Allen who was paired with a scarab and has been secretly altering history to help the Reach. Well, sort of. I'm thinking that while the Titans were hurtling through Quantum Space, time was passing much faster for everyone else, Earth was nearly destroyed by the Reach, and the Flash decided to run back to before the invasion to warn the Justice League. But what he didn't realize was that the time remnant followed him back, and actually caused the attack to happen sooner. This started a chain reaction where the two Flashes kept chasing each other back and forth through time until the Speed Force nearly collapsed in on itself. As a defense mechanism, the Speed Force strengthened it's barriers, making the timeline much less malleable and essentially setting the new Flashpoint reality in stone. 

For the Titans, this entire process happened while they were in Quantum Space, which shielded them from the timeline changes. But their home is different, there are alternate versions of them walking around, and the Reach is still a threat. Ultimately, we are able to defeat them with an advanced computer virus that deactivates the scarabs, but our world is in shambles. No matter how hard the Flash tries, the Speed Force blocks him from making any further changes to history. Luckily, they are able to find a work-around in the form of a new modified Quantum Warp Drive, but when they attempt to use it to jump back in time, it explodes and fractures reality into multiple parallel timelines, during which we get to visit many different versions of Earth. 

Ultimately, Barry Allen is able to discover a backup of their original timeline frozen in a pocket dimension within the Speed Force, which he is able to restore as the Prime Universe. They still have a copy of the Reach virus, which they are able to use to win the war fairly quickly, thus saving Earth. Finally, things seem to return to normal. 

I am also envisioning a few title changes to the RP as the plot changes and morphs:

Teen Titans: Lost Galaxy
Teen Titans: Flashpoint
Teen Titans: Crisis on Infinite Earths
Teen Titans: Rebirth
Teen Titans: The Judas Contract
(then a five year time jump)
The New Titans: Infinite Crisis
The New Titans: Lantern Wars
The New Titans: Blackest Night
The New Titans: Brightest Day
The New Titans: Final Crisis

Just some ideas, this is all subject to change, lol.