This story starts at a point after Season 4. Twilight, the new Princess of Friendship, takes her new role seriously, and throws her entire being into becoming the best Diplomat anypony has every seen, with her friends included.
Almost in response to this, Twilight discovers a gateway in the basement of her new crystal tree palace. Upon study, she finds that it can take her to other dimensions. After much deliberation and preparation, Twilight, having already used a gateway to another world before, so being the foremost expert on such things, uses the gateway along with her friends to cautiously and carefully open communications with other races in order to spread the joy of harmony and friendship, as well as open routes for trade and such.
Cue Twilight entering the gate with her friends. Upon appearance, it's the middle of the night. Almost immediately, Twilight detects a powerful magic signature that's distressed and in great pain. The Elements of Harmony go to the source and find a baby on a doorstep under an enforced magical sleep, with tons of magical compulsions, and other such things attached, and on top of that the Baby hasn't eaten in at least a day.
Horrified, Twilight reads the note from Dumbledore to Petunia, and becomes outraged on such a thing happening. Twilight decides to cut their losses before this Dumbledore attempts to use Equestria to his advantage as well and rescue the poor babe, severing all compulsions, and taking the babe back to Equestria, where between some prominent Unicorn doctors and Celestia herself, they get rid of the soul leech in his forehead, allowing the scar to heal to a thin white line, barely noticeable unless you look really closely.
Twilight takes responsibility for the infant, and between going out periodically to set up more beneficial treaties and trade agreements with other species out there, little Harry Potter grows up with a loving mom, five adoring aunts, a dragon for an older brother, and three grandmas, two of them honorary, what between Twilight Velvet, Celestia, and Luna.
Twilight, when Harry comes of age and having studied him quite intensively by this point, gives Harry, who lacks a focus like a horn that the Unicorns have, a circlet with a lavender colored focal gem for Harry to cast magic, and attempts to teach Harry the magic of Unicorns, to great success. Harry, not being limited to just Unicorn magic, also attempts to learn Pegasi and Earth pony magic from RD and AJ mainly, but the other mane 6 as well, gaining the ability to manipulate clouds and weather, as well as hover and fly slowly through the air, as well as growing plants and learning to buck apples and getting really strong legs from it from Applejack. He takes to it like a natural. Many hijinks ensue, especially between the two brothers as well, when Harry reveals a mischievous streak, makes friends with the CMC, goes pranking with Rainbow and Pinkie, and other such events happen.
Eventually, upon reaching his 15th year of age, Harry starts to wonder about his origins and questions Twilight about where he came from. Twilight, knowing this day would come, tells him as much as she can and leaves nothing out. Harry, understandably, wants to know why he was left on a doorstep, who his original parents were, and other such things. Twilight, having already dealt with Spike's need to discover his origins already, accepts this and helps Harry prepare for his foray back into his old world, giving him a number of enchanted items for his protection, as well as a hidden dead switch that would teleport him straight back home if he's ever knocked out with malicious intent.
At this point, the story could go any number of ways. Whether Harry goes alone or with the CMC or the mane 6 or just some of the mane 6 depends on if Harry convinces Twilight and the others he needs to go alone or what have you.
Upon going, Harry could pop out right back in the street where Twilight showed up and start his quest to find answers, using his ability to sense magic to find Diagon Alley and such.
Or… in a twist of fate, Harry could appear in the middle of the great hall after the Goblet of Fire spits his name out, the magic of it drawing him in mid transit to the Goblet, upon which then he can get some answers. Good thing the mental arts are something of a specialty for Twilight so no snooping in his head for old meddling headmasters.
At that point, well… you have an Equestrian trained Harry Potter in the Triwizard tournament. Go nuts!
No seriously, go nuts. I hate reading stories where canon is followed to the letter despite an extra character being thrown in, or something monumental to change how a character acts happens. I believe in the butterfly effect and chaos theory, so stories where something fundamental happens but it changes nothing just ticks me off. I've already read the canon story, thanks. Give me something new.