The way it happens is:
"Hey, what's this?" Sirius tilts the flask, watches the water within swirl.
Lily lurches up, eyes wide and hand reaching. "No - !"
Slow-motion, Sirius jerks at Lily's cry – the flask upends – the water falls – onto baby Harry's upturned head –
Lily palms her face. "This is why you don't play with rare, complex Potions ingredients," she groans. "You get to explain to James why his son is now his daughter."
The backstory, long version:
Severus was the better potions brewer, but Slughorn always liked Lily more, which was why even years out of Hogwarts he still called upon her to collaborate on his newer experiments. (Besides, there were all these rumours about what Severus was getting up to these days, and Slughorn and Dumbledore and Lily all thought it best to keep him out of the cutting edge loop.)
(Unbeknownst to them, Severus Snape was cutting his own edge, sharp, and careless enough with it to bleed and not care about the stain.)
This was why the Potter residence was rather perpetually overrun with esoteric potions materials and equipment (gifted and borrowed, respectively, as Slughorn was generous with his favourites); this was why there was water from a cursed spring waiting to be added to a Tiresias potion experimental variant; this was why Sirius had been banned from Lily's workroom for the past five months which Lily probably should have realized was a mistake, because if nothing else so encouraged the mutt as a locked door it was a forbidden locked door; this was why Sirius was an idiot and Lily was a bigger idiot for letting him through her front door and James the biggest idiot of them all for inviting him over; this was why Harry's wild black hair had suddenly morphed into wavy, silky red.
The backstory, short version:
"What do you mean, Jusenkyou?-!" James roars.
No one panics because:
"Of course there's a cure," Lily scoffs. She's busy braiding ribbons into Harry's flame red curls. "All Harry needs is a dousing with water from the spring of the drowned boy" ("That is such a creepily named spring," Sirius whispers) "And he won't turn into a girl every time he gets covered with cold water. The quick fix until I can get in touch with Horace's" ("It is so creepy that she calls him Horace," James whispers back) "Jusenkyou supplier is to splash Harry with hot water. The curse is triggered by water of varying temperatures, it's rather ingenious actually –"
This is about the point where both men tune Lily out.
No one else knows because:
Lily never gets the chance to call Slughorn up and relate the whole ridiculous tale; that night Dumbledore comes to them with the warning that they must go into hiding, now, it is urgent – and they must not be in contact with anyone, not even their Secret Keeper, not even their best friends – and then they are dead, and Sirius as good as for all the use he is – and Harry has become the Boy-Who-Lived, only he's not even always a boy.
Ten years later:
When the Hogwarts letter comes, the Dursleys are relieved. Finally the freak is no longer their problem.
Maybe if Harry had been – well, normal, even a little bit – maybe then, with something to work with, they might try to fix him. Get the freakishness out. But it's been obvious since the first time Petunia gave the thing a bath that there was no redeeming Lily's child.
"Good riddance," Vernon mumbles, shoving Harry out the car door outside the train station, September 1st. From the back seat Dudley glares with his piggish eyes. Petunia in the passenger seat doesn't deign to glance farewell.
Harry inhales, deep, straightening up, arranging trunk and owl cage for ease of travel (Hedwig was a gift from Hagrid, the larger-than-life guide to Diagon Alley), steps forward, into a new world.