basically, i'll try to keep this brief about patronuses. specifically,
the first thing we know is that, for animagi, the patronus matches their animagus form. [purely theoretically, yes, an animagus's patronus could probably change, but as far as we know, we don't actually have any confirmation of that]
since James's patronus matches his animagus form, that means it's his original patronus.
his patronus never changed.
Rowling also said that this [Potterās Patronuses] wasn't a coincidence, and that patronuses can change because of love.
Chely: James patronus is a stag and lilys a doe is that a coincidence?
J.K. Rowling: No, the Patronus often mutates to take the image of the love of one's life (because they so often become the 'happy thought' that generates a Patronus).
30 July 2007 Bloomsbury live chat with J.K. Rowling
accordingly, Lily's patronus changed because of James's patronus.
what else do we know about the patronus itself?
the patronus charm is an extremely advanced spell, far beyond O.W.L. level. and by "far beyond", i mean it simply isn't taught at school. [the only reason Harry was taught it at all was because the dementors had been brought into Hogwarts. that's literally the whole reason]
we also know that the Order of the Phoenix used patronuses extensively as a means of communication, which explains why, from what we know, all of its members were able to produce one.
we also know that NONE of the Death Eaters use patronuses.
so, based on all of this, Lily and James would have learned to produce their patronuses after joining the Order of the Phoenix, because it wasn't part of the Hogwarts curriculum.
at the same time, Severus most likely couldn't produce a patronus at all before the age of 21. again, because it wasn't taught at school, it doesn't seem like the sort of magic that would have interested him, and, as a Death Eater, he obviously wouldn't have had any reason to practise it.
after joining the Order, however, he would have learned the spell.
there's one more thing we know about patronuses. in the original writing that was first published on Pottermore, it's stated that a patronus may change because of powerful emotions, such as falling in love, bereavement, or a profound shift in a person's character.
The form of a Patronus may change during the course of a witch or wizardās life. Instances have been known of the form of the Patronus transforming due to bereavement, falling in love or profound shifts in a personās character.
when Severus casts a patronus for the first time, the emotion he's experiencing is bereavement. that word wasn't included there by accident.
// and no, i absolutely do not believe that the doe was Snape's original patronus, because it suits him as poorly as it possibly could.
imho, headcanon whatever you like, but i'll never agree with that interpretation.