First of all, there can be more than one Slayer. There’s about a dozen of them active at any one time. Old ones do not have to die to activate new ones, which loses some of the “doomed to die young” angst, but a Slayer’s life is still brutally short. There have been times when the entire Slayer line has been wiped out, and it was decades before the Watcher’s Council was able to build their strength back up.
Changes, broken down by season:
Season One:
Season One is a full season, not a stunted mid-season replacement. Buffy is still a newly-transfered sophmore, but she doesn’t know she’s a Slayer in the first episode.
Buffy doesn’t die. What with there already being other Slayers, her death isn’t needed as an excuse to bring in Kendra, and later Faith.
Season Two:
Get rid of Acathla: he was too much like the Judge. Since he’s basically just a McGuffin, have Angel trying to destroy the world by opening the Hellmouth, just like the Master was trying to do. Also cuts down on the “How many ancient apocalyptic artifacts of evil can be in one town?” factor. And, since it’s a Hellmouth, Angel ending up in Hell makes more sense. Since it’s a permanent gateway, Angel finding his way back through it next season makes more sense, too.
Season three:
Xander loses an eye now, not later. It happens during graduation, and it happens because Buffy is forced to choose between him and something else, and chooses something else. Possibly Angel. Feeling betrayed, he leaves Sunnydale to join the army, which accepts him despite the fact that having only one eye probably disqualifies you from military service.
Season four:
Because of the incredibly public events* of last season, it is now generally known among the inhabitants of Sunnydale that Buffy is a superhero. Among other things, she receives aid from the police, although the police are still mind-bogglingly incompetant.
Oz still leaves the show so Willow can fall in love with Tara, but he comes back next year with his own spin-off.
In the last act of “Hush”, Buffy finally meets one of those mysterious commandos face-to-face: it’s not Riley, it’s Bitter, One-Eyed Xander! Over the course of the season, he re-learns to trust Buffy and the Scoobies, falls in love with Anya, reconciles with Willow, recognizes the corruption of the Iniative, and gives Spike lots and lots of electric shocks. (Bitter One-Eyed Xander is in the show from the first episode of the season, but he avoids earlier contact with Buffy. Because of the Bitterness.)
All instances of Buffy boinking Riley are replaced with Buffy boinking her hunky contact in the Sunnydale PD. Until Xander gets back to his old self, he serves as the “sarcastic observational humor guy” so needed to make the chemistry of the show work. Once Xander gets over the Bitterness, PD guy buys the farm, to clear the way for later Spuffiness. Maybe he survives into next season. (Unless his character turns out really cool: then he gets his own spin-off. Everyone gets a spin-off. And all the spin-offs are really, really good.)
Season Five: Following the failure of the Initiative, but faced with rising supernatural threats, the government contacts Buffy directly, and offers to support her in her efforts against demons and their ilk. Basically, this is an alternative information source to Giles and his endless supply of musty old books. Plus, they’re good for the occasional loan of a rocket launcher, and they also keep Useful, Heavily Armed Xander supplyed with bullets and hand grenades. After Joyce kicks it, they pay Buffy a modest stipend to keep her out of the Double Meat Palace. They buy up the land Sunnydale High was on and the surrounding neighbor hood, tear it all down, and build a small military complex on it, including a twenty ton steel-reinforced concrete block sitting right on top of the Hellmouth.
Glory is still Glory, but instead of occasionally turning into Ben, she occasionally turns into a forty-story tall writhing pillar of flesh.
Giles leaves the show at the end of the season, but goes directly into Ripper, where he cleans up the Watcher’s Council and transforms it from a corrupt beauracracy into a lean, mean, evil-fightin’ machine.
Buffy doesn’t die.
Season Six: Having killed a God, Buffy comes to personal attention of the Powers that Be, and becomes involved with extra-planar intrigues and the universal war between good and evil. She doesn’t go traipsing off to Pylea every other episode; more like more gods (hell and otherwise) become interested in Sunnydale.
Communication and travel with these beings is fascilitated by Dawn’s mystical Key powers.
At this point, Buffy is essentially a warlord in a vast invisibile war, and Sunnydale is her territory. Other “warlords” such as Angel in L.A., and Oz up in Oregon (or wherever the hell his show is set. Probably somewhere woodsy.), vie for influence and power with each other. Obviously, Angel and Oz are allies with Buffy, but other territories are held by powerful vampires, demon kingpins, etc. The “big bad” requires Buffy to be diplomatic and politic, not just punch things real hard. Of course, there’re constant turf wars and skirmishes, so the show still has plenty action. This has been building up since they left high school, but this season it really comes to the fore of the plot.
Xander doesn’t leave Anya at the altar, but he still leaves her. Then she leaves the show. She does not get a spin-off.
The Geek Trio is still around, but Warren doesn’t make it past mid-season. After meddling in the Slayer’s affairs several times, Willow (who has been delving a little too deeply into the dark magics) suggests harsher reprisals, but Buffy holds back. Warren gets Tara killed, Willow kills warren and breaks with Buffy, blaming her for being “too soft.” She decides to set herself up as a warlord in her own right, but turns to darker and darker powers to secure her territory (which is some subsection of Sunnydale) Eventually, she goes entirely Dark Willow, challenges Buffy for control of Sunnydale, and gets an ass whupping from Guest-Starring Giles. The finale plays out more or less the same, except instead of the Random Evil Temple that Willow calls out of the Earth, she busts in on the military base, whups soldier ass, then gets the breaky crayon story and repents of her wickedness.
Season Seven: Basically the same story line, but better done. Obviously, there’s no re-built highschool. Willow’s attempt to open the Hellmouth last season damaged the seal that had been put on it, and all sorts of bizarre phenomenon happen to the army guys. Eventually, they all get turned into bringers except for a few minor characters who escape and join up with Buffy. Vampires and other demons are drawn to the awakening Hellmouth, turning the bunker into the First Evil’s fortress. The season finale takes place here, probably plays out much like season four, only bigger. By season’s end, Sunnydale has lost all pretense of normalcy, and is basically an armed camp of refugees surrounded by rampaging demons.
Since British shows never run more than a series or two, Giles rejoins the cast after Ripper ends with the dynamiting of the Watcher’s Council.
Instead of Slayers in Training, all the other Slayers world-wide start gathering in Sunnydale. Some of them are still in training (but semi-super powered) others have been fighting longer than Buffy. So they’re actually useful in a fight. Also, they’re not as likely to follow Buffy’s orders, leading to further plot convolution. Kennedy is played by someone else, and is better written. And Xander gets a Slayer too. Let’s make this as Freudian as possible. Oo, Xander’s Slayer girlfriend is the girl who only speaks Chinese! Much hilarity ensues.
Caleb is still in it, and think how bad ass he would be if he’d just slaughtered two full-fledged Slayers!
Buffy is essentially a military leader, with multiple squads of trained warriors, and acts like one: recon, feints, tactics, etc. She has the Slayers, Spike, Faith (after she does her stint on Angel, and before she gets her own spin-off) the Scooby gang, the more competent Sunnydale cops, and a bunch military guys left over from the bunker over the Hellmouth.
Buffy dies.
*I italicize this because I feel this is the point where the show first started losing it: when the mayor turns into a giant snake and attacks a graudation ceremony, of which all the participating students are expecting and fully prepared, and the next season everything goes back to normal. I always wanted the show to deal with the effects of more and more people knowing about the supernatural, and that theme informs most of the rest of my ideas.
Now ask me how I’d change the original Star Wars trilogy.