So Chell is free now. What would she do? I suspect she would be the kind of loner who would live in the forest somewhere and find an abandoned cabin that she'd live in. In the beginning of the game, the player don't know where Chell is. GLaDOS decided to not track her.
Then the war on the outside gets dangerously near the Aperture laboratories. GLaDOS sends up a few robots to investigate. An encounter with Black Mesa follows and she thinks that they is responsible for the explosions nearby, until she sees the aliens. When she realises she and the entire lab is in danger if the aliens wins (she barely consider Black Mesa a threat), she carefully asks a group of Black Mesa soldiers (through her robots) if they should work together to fight the aliens (at first she was about to attack them and interrogate them about everything that has happened on the outside, but has to change her mind and realize they know much more than she does about the situation). She also sends out Atlas and P-body to find Chell, hoping that she can help.
Out in space, aliens in a spaceship spots Wheatley on the radar and goes and gets him (and the space core). Stupid as he is he didn't even care to remember where the laboratiories is, but he reveals enough information to make the aliens start looking for the lab, thinking there's useful technology for them there. GLaDOS realize this when one of her scouting robots overhear some kind of alien asking Black Mesa staff that it caught about where the Aperture lab is. GLaDOS then sends out many more robots looking for Chell and starts a massive weapons research program, which Black Mesa realize that they have to contribute to if they want to survive.
Gordon Freeman (FYI, Wheatley's internal clock was waaay off, it hasn't actually been *that* long time since the first game) is hunted by aliens, and drops a gravity gun. This happens to be near Chell. And so Chell encounters aliens for the first time when she hears explosions and goes to look, and she then spots the gravity gun. Gordon, who was about to be caught, now hears much louder blasts (Chell is a fast learner) which also distracts the alien that chased him, who he then beats. He then sees a large alien ship crash, and Chell shows up right after that with the gravity gun in her hands. She realize that it's probably his gun, and untalkative as she is she just holds it up for him with an expression that says "is this yours?". He asks her if she wants to help save the world, and says she can keep the gravity gun if she helps him, and she agrees with a silent nod.
Then the group encounters some of GLaDOS' robots which are fighting aliens a that time. Suddenly Atlas and P-body shows up in the fight (the player didn't see them at first), but only Gordon sees them at first. Then P-body spots Chell, and tells Atlas that she is there, which almost gets the two robots destroyed as they're distracted. They all quickly start fighting side by side when Gordon realize that they are "friends", sort of. Then GLaDOS asks through the robots about Chell's new friends, so Gordon Freeman introduces himself. Then GLaDOS reveals that she knows that last name, and asks about his family. Then she tells a short story about a relative to him which she got from the Aperture laboratory databases (this one was something he already had heard about, but she responds with that she got more stories for later). She also tells them some of the info about the situation she got from Black Mesa.
Now GLaDOS realizes that the aliens found out where the Aperture labs is. She decides to initiate a backup plan (literally making backups of herself and important data, so she could establish entire replicas of the Aperture Science labs elsewhere) while also trying to fight the aliens. Then a robot sent out by GLaDOS meets up with them on the way back to Aperture, which gives Chell a new ASHPD (occassionally Chell got to borrow an ASHPD from Atlas or P-body until now - and no, no moon in sight). But this one was modified to be both a gravity gun and a portal device (so it can throw things just like the gravity gun; built using information from Black Mesa). Atlas and P-body also gets new upgraded ASHPD:s, and the robots also gets more raw firepower. GLaDOS hesitates to give Gordon any new weapons, but gives him more ammo.
The group arrives at Aperture at the same time as the aliens.
Edit: Black Mesa soldiers also arrive, and Black Mesa and GLaDOS are still considering themselves as rivals to each other, but works together here at first. Once most of the aliens have been fought off, Black Mesa starts attacking GLaDOS, and she fights back. Your group is left alone by both of these, but there's plenty of crossfire to avoid, while fighting the remaining aliens. Chell and her group have a hard time to decide if they should help GLaDOS or not.
To be continued. I'm still thinking about how the story could go. I have no idea where to go from here.
FYI: Old Portal players might prefer to play less violent puzzle solving straight through the game, just like in the Portal games (I'm in this group). The sequel should let the player play that way. But Half Life players might also want to play more "Half-Lifey" through the entire game, and that should probably be possible too. Maybe the player shoudl be able to decide between a path away from aliens that require puzzle solving to get through, while also being able to face the aliens directly in the levels. Or there's just one level, but several alternative ways to get through it, including both pure fighting and pure puzzle solving.