Just a blurb of an idea that's been sitting on my computer for a while... I might try and do it properly at some point. Harry Potter and Doctor Who do not belong to me, and I make no money. La di da.
Human Plus
Lily Potter knows her son is a bit different.
Oh, it's not the seemingly bizarre victory of her Muggle-born genetics over James' pure-blood ancestry that does it (though really, James aside, everyone exclaims over Harry's tufts of ginger hair and looks at her suspiciously for it), nor is it the ridiculous amount of accidental magic her son performs as an infant (something, both parents note, that Dumbledore watches carefully with hooded eyes). James has ginger relatives and while he's better with a wand than her, they are both magically powerful, so what's the big deal?
For all his magical activity, Harry's a very quiet child, and she can tell already, very, very intelligent. It's in the way he studies things; babies are notorious for sticking things in their mouths, but Harry grabs them, stares them down, turns them over in his baby hands, sniffs them, and licks them instead. It should be more dangerous that it is, because Lily's working on her Masters in Potion Brewing certification and however careful she is to clean up after herself, ingredients inevitably end up in inconvenient places in their Godric's Hollow home; but her son manages to acquaint himself with almost everything without causing harm to himself, a skill some of Slughorn's seventh years at Hogwarts never learned.
It's like he knows if I do that, bad things happen without it having to happen to teach him.
That's not normal, Lily realizes, but she and James agree, it's a good not normal.
(So there's no reason to send the Doctor a message, no need to ask if carrying a baby on the TARDIS may have had an effect, no excuse to pull him away from whatever planet or starship or generic civilization he's saving to sooth their slightly frazzled nerves when Dumbledore tells them about a prophecy and a power the dark lord knows not…)
James is dead in the hallway and Lily is dead in the nursery. Harry pokes his head up in his crib inquisitively and finds the business end of Voldemort's wand in his face. A more human form of Tom Riddle would've found it a bit amusing how the baby's green eyes go cross-eyed for a second looking at it, before the boy looks up, peering through a ginger mop at the wizard about to kill him.
(Potters have never been ginger, but Harry is a lot of things Potters have never been before.)
Voldemort says Avada Kedavra, a second flash of sinister green light illuminates the room, and in the following second of stillness before he checks that his prophesied enemy is dead, Voldemort is blasted from his feet and his body by an explosion of brilliant gold energy.
It is some hours before anyone is brave enough to come near the wreckage of the house, by which time the regenerative energy has done its work and settled down (Harry is still so very young, and there's not much to do). Somehow in the chaos no one questions the inky-black untidy Potter hair, obvious though the rest of the change is buried in a toddler's body.
(Perhaps they just see green eyes and a lightning-shaped scar, where necrotic energy and regenerative energy fought and cancelled out. Later, Petunia Dursley will notice that some of Harry's fringe still grows out ginger, and she will not approve at all.)
Harry knows he's a bit different from everyone.
It's not that he's a freak, whatever his relatives try to tell people – or if he is, he refuses to see it as a bad thing. (His relatives call themselves normal, after all.) He thinks maybe he's just a bit smarter than most people, but on Privet Drive, where being bigger and meaner is the most important thing, that's not much of a claim.
Surrey Primary doesn't have accelerated courses, nor would his relatives allow him to take them if it did, so Harry contents himself with making numbers games out of his homework assignments and dodging Dudley's gang. His primary school teachers never notice that he makes precisely the same percentage grade in all of his classes. It's only slightly less boring than if he just did everything right, but it causes less problems at his relatives' house. (It's never been home.) They also never notice when he finishes his tests in five minutes and spends the rest doodling, drawing patterns of perfect circles and ripping them up because they mean something he doesn't know how to say. Harry never makes a deal out of his boredom, though, or the ease of the material. It's not all that impressive, he thinks, since his teachers don't expect any of their students to be smart.
(One day, avoiding Dudley, he dashes across the way and finds himself in the library at Stonewall Secondary, and disturbs a student struggling with a difficult calculus problem. Harry glances over the relevant section in the book and solves it in ten seconds, and the student has him thrown out of the library.)
It's not his brains that indicate to Harry that he's different. The truth is, he knows things; it's not always or even often, but like a fragment of a dream, he's at a crossroads, and one way this and another way that, and this is bad this is okay don't go here, and rarely, very rarely, the roads all combine to spell out this-will-come-to-pass.
(He remembers green and gold light when he tries to recall the real way his parents died – green and gold light and terrifying certainty. People aren't fixed like that in a simple drunken car crash, and every time they tell that lie, Harry hates the Dursleys that much more.)