This series of stories takes place in an alternate universe. In this version of events, John Sheppard was dishonorably discharged after Afghanistan, so he never went to Antarctica and never joined the expedition. Rodney was slated to be on the expedition, but a few days before departure the Trust got hold of him and implanted him with a Goa'uld. Carter figured it out just in time, and when the expedition left Rodney was in confinement waiting for removal of the parasite. The removal didn't go so well, and it took Rodney a while to recover -- so when the Daedalus departed ten months later, Rodney still wasn't cleared to go with them.
(Jim Ellison and Blair Sandburg also had some trouble with the Trust, leading them to join the expedition to get away from their pursuers. If you want to know more about that, you should be reading the stories instead of the summary.)
Needing someone with a strong ATA gene, Elizabeth persuaded mini-Jack to join the expedition (that's the teenage-appearing Asgard-made clone of Jack O'Neill, with all of O'Neill's memories up to early seventh season). In this universe, the Asgard's artificial aging left mini-Jack appearing more like seventeen rather than the fourteen-year-old from canon, and Elizabeth managed to swing him a Lieutenancy for the expedition.
Their first day in Atlantis, O'Neill rescued Sumner halfway through the Wraith feeding. They ended up with the prematurely-aged Sumner in nominal command and O'Neill doing all the real work. They had many of the same adventures as our team in canon for the first season: Zelenka was killed by the nanovirus; Kavanagh was killed by the 10,000-year-old Wraith; Grodin survived. Their extragalactic dialing crystal was damaged by the mist-aliens, so even though they got the ZPM from Dagan, they still weren't able to dial Earth. They got the ZPM installed and the shield up just in time to stop the siege (although some Wraith did make it into the city).
The Daedalus arrived in the middle of the siege, unwarned about the Wraith and without a ZPM to bolster their shields. The ship took heavy damage, and O'Neill left the city in a cloaked jumper to try to rescue some survivors. On the ship, he met up with a Wraith and got aged a few more years (and also had one pupil blown out like Ford's), but he got back to Atlantis with a crippled Caldwell and made it through the enzyme withdrawal. They faked the destruction of Atlantis and kept a low profile through the equivalent of second season. Caldwell was in command from a wheelchair now, Sumner having died of his Wraith injuries; O'Neill got a field promotion to Captain. Then the Genii invaded, took over the city, and put everyone in the brig.
Somewhere in late second or early third season, a canon-consistent Sheppard touched a weird device and ended up getting switched with alt-Sheppard. Sheppard went to the SGC, and with Rodney's help managed a one-time power source to dial Atlantis. They put together a hasty strike force to go through the gate and save the city from the Genii. Then they went to the planet with the device Sheppard had touched, and so got him back to his own universe. (Cross Product)
Alt-Sheppard was torn away from his surfing to an alternate universe, and then popped back to his own universe only to find he was now stuck in the wrong galaxy. Elizabeth offered him a job as a (civilian) Gateship pilot. He and Rodney are together and are on a team with Ellison and Sandburg. Adventures ensue. (John settles into Atlantis and joins the team: Cross Multiplication; they deal with their Genii prisoners: Cross Training; they undertake some salvage operations: Cross Country; etc. . .)
Cross Talk, Crossing Paths, and Crossed Wires are some stories which take place shortly after and show some different perspectives. Future stories will include the one where they find Ronon and the one where O'Neill gets captured by mermaids . . . um, yeah.
Geeze, that's a long summary.