(Excerpts of the following were taken verbatim from suggestions made by those listed above.)
And now, the alternate universe:
Sometime after the fall of the Soviet Union, a highly secretive, multinational intelligence organization was formed, called the Atlantis Cooperative. The agency combed the ranks of the intelligence community, military, and universities around the globe looking for the best of the best to serve its member nations. Along the way, the Coop has dealt with bad guys in all shapes and sizes, and been key in the security of people around the globe.
John Sheppard: Served in the Air Force as a helicopter pilot after graduating from college on an ROTC scholarship, but was soon lured by the NSA as a cryptographer. While he is gifted in mathematics, with a surprising adeptness for field work, he's not so talented working with his superiors, and making his boss mad one too many times got him shipped off to the Atlantis Cooperative. There he stayed fairly low under the radar, until running into Elizabeth Weir and getting roped into field work.
Elizabeth Weir: A political science major with a few languages learned on the side. Elizabeth was originally recruited by the CIA in a time when fluency in Russian was still a major asset for the agency, but made her way over to the Atlantis Cooperative after the breakup of the Soviet Union. She worked as a translator and analyst for a while, but Jack O'Neill discovered her and pulled her into the field as his partner. In the meantime, he gave her a code name she hated and ended up having an affair with her before his retirement. Shortly after Jack retired, however, Elizabeth's second partner, Marshall Sumner, died on a mission. Before meeting Sheppard, she worked alone for a few years. Gifted in languages, she can pose as a native of just about anywhere in Europe (and often does).
Jack O'Neill: Former agent who trained Elizabeth Weir. He's retired in Minnesota, but keeps getting phone calls from the home store, wanting his input. He'd throw his cell phone in the lake, but that might kill the fish.
Sam Carter, Rodney McKay, Radek Zelenka, and Lee: Research and development. Just because Carter and McKay can't stand each other doesn't mean they shouldn't have to share control of a department, right? McKay may have broader range in the most technical sense, but Carter is a natural in high-tension situations, whether in or out of the lab. The operations and intelligence departments are always trying to get her to transfer over, and Rodney has his pen ready to sign the transfer papers. Zelenka and Lee are two top engineers in their department.
Daniel Jackson: A linguist recruited from the NSA, and the best now employed by the Atlantis Cooperative. Not the best field agent, but he's learning through trial and error. He possesses remarkable luck, and has survived things that make more experienced agents cry. Generally, he works as an analyst in the signals department, seeking info that will keep the agency one step ahead of its enemies. He's also an expert in the customs of many, many cultures, especially in the Middle East. (Works closely with Weir when she's not in the field, might have a bit of a thing for her.)
Teal'c: No one quite knows where this imposing man got his name, his tattoo, or his wardrobe. A resident of a tropical island, he's an arms dealer who seems not to care much about sides in the battles that the Atlantis Coop fights – except when the right people approach him. He had a good relationship with Jack O'Neill and thus with his former partner, Elizabeth Weir. Weir and Sheppard drop by to exchange briefcases of money for information, while Teal'c sips at fruity drinks, wears Hawaiian shirts, makes fun of Sheppard, and compliments Weir's designer shoes.
Steven Caldwell and Hank Landry: Directors of Operations and Intelligence, respectively. The two vie for power in the agency, as each has his own opinion on which department should handle which mission.
Cameron Mitchell: A newly recruited black-ops commando and former Air Force pilot, who was honorably discharged after barely making it out of a burning cockpit alive. It was no real surprise that the intelligence community came to call after his recovery.
George Hammond: Caldwell and Landry's superior, who answers to a mysterious group of higher-ups referred to as the senior partners.
Carson Beckett and Janet Fraiser: Two brilliant doctors whose work is vastly different. While Beckett deals mostly with injured agents, Fraiser works closely with research and development as well as operations as a forensic investigator.
Jonas Quinn: A recent recruit who's still trying to figure out what his place in the agency is. He has the misfortune of being good at everything, so no one quite knows where to put him.
Paul Davis and Chekov: Representatives of their country to the Atlantis Cooperative. As the group is multinational, there's a great deal of political wrangling to get the agency's efforts focused on various issues around the globe. Davis, Chekov, and their counterparts from other nations spend a great deal of time lobbying the senior partners and trying to influence the field agents, analysts, and researchers to sway them to their countries' needs.
Characters who have been left out are ones who, in all likelihood, were simply difficult to place in the AU. However, the fact that they don't appear here doesn't mean they can't appear in the AU. If you've got a good place to put them, suggest it. Or better yet, write a fic.
As for the enemies, only two have really been nailed down at present, though there are many, many other possibilities with the Wraith, the Replicators, and the Ori. At this juncture, the NID serves much the same purpose as it does in canon. They're rogue intelligence and corporate espionage. The Goa'uld, on the other hand, are international organized crime, from a wide variety of cultures, and dealing in drugs and people and all manner of crimes against humanity. The thing that's helping the Atlantis Cooperative keep them at bay is their own internal turf wars.
Again, if you have suggestions, please feel free to share.