thank you very much for the ask, anon!
although i regret to say that i'm going to start the answer to it by being very pedantic...
the idea that masteries are something which exist in the wizarding world is complete fanon.
they have emerged as a trope due to a reading of the phrase "potions master" which does make perfect sense outside of the cultural context in which the books were written - by which i mean that it makes readers unfamiliar with the culturally-specific meaning of this bit of language think of masters degrees or other high-level qualifications - but which is nonetheless incorrect within context.
"master" [and the feminine equivalent, "mistress"] is just an alternative term in british english for "teacher". it doesn't imply anything about a level of qualification. "potions master" and "potions teacher" are synonyms.
the term is archaic - british people nowadays would exclusively say "teacher" - and it's very class-specific, in that it would have particularly been used to describe teachers in elite schools, whether fee-paying private schools or grammar schools [state schools which are academically selective].
as a result, it turns up in lots of the children's literature written before c.1980 - especially in boarding-school stories like malory towers and the worst witch which are explicit influences on the harry potter series. it's used in the text - especially in the earlier books - as part of worldbuilding which generally seeks to make the wizarding world feel whimsical by virtue of being very old-fashioned, which things like the fact that the most advanced technology wizards use is the radio and the steam train also hammer home.
that snape is the only teacher referred to as a master is connected to these genre conventions. because snape is so important to the full arc of the story, he's the teacher we spend the most time in the classroom with throughout the six books in which harry's at school. and he's therefore the teacher who - in the first few books - best fits a children's literature archetype which we would expect to find in any twentieth-century school story [with a magical setting or not] - the hated schoolmaster who is horrible to the child-protagonist and who every child reading can't wait to see get their comeuppance.
so snape is a potions master because he teaches potions. nothing more than that.
but that doesn't mean that it's not worth thinking about his training...
clearly, higher education of the type most of us are familiar with doesn't exist in wizarding britain - nor, i suspect, in wizarding europe more broadly.
and this makes perfect sense - not only because the magical population is so small but because the divergence of the magical and muggle worlds in 1689 takes place well before universities and university-level education look like anything a modern student might recognise. a seventeenth-century university education was still broadly generalist and aimed at trainee clergy, and careers which we would nowadays expect to require a degree - such as law, finance, medicine, science, and engineering - were generally taught by apprenticeship.
this is clearly how things continue to function in the wizarding world of the 1990s, since we know from order of the phoenix that healers are taught by apprenticeship [and, indeed, that hogwarts graduates all go straight into the workforce after they leave school].
potions - since it's analogous to chemistry - is nonetheless understood in-world as an academic discipline. but this doesn't mean - within the post-school educational structures we can suppose the wizarding world has - that it's a discipline in which one needs specific formal training in order to acquire a right to teach or publish about it.
the seventeenth century was a period - especially in britain - marked by a great expansion of scientific inquiry. this was - by our contemporary understanding of academic science - amateur. scientists wouldn't have been expected to have doctorates, to work at universities, or even to have attended them, and their experiments were often self-funded by personal wealth or dependent on a patron. the circles [often international] in which they debated, demonstrated, and reviewed theories and inventions were social ones - the gatekeeping line was class [with the level of education - and, primarily, of literacy - that this implied], rather than level of education itself.
these social circles often had a certain level of official standing - by which i mean they became, during the period, the learned societies, the most famous of which is probably the royal society. membership [or fellowship] of the learned societies requires a demonstration of some sort of contribution to the discipline they relate to - which means that the vast majority of contemporary fellows of such societies are university-based academics. but this wouldn't have been the case in 1689.
and we know that the wizarding world has its own equivalent of learned societies, because slughorn mentions one in half-blood prince - the most extraordinary society of potioneers.
which is to say, snape is probably a member of this society. he may very well publish papers in academic journals connected to the subject [as dumbledore does in transfiguration today], and he undoubtedly has a reputation among the wizarding world's men- and women-of-letters. but he doesn't need to have any formal post-hogwarts qualification in order for him to have acquired this reputation.
so what do i think he's doing between 1978 and 1981?
well... he's a death eater.
my theory has always been that snape comes to voldemort's attention - via lucius malfoy - because of his potions skills. the dark lord's operation would have needed potions - poisons to bump off enemies, healing potions because wanted criminals can't just turn up at st mungo's, potions to trade on the black market [as aberforth dumbledore tells us the death eaters do during deathly hallows], and so on - and voldemort would want to keep the production of these potions in-house, rather than risk hiring a private brewer [even a shady one] who might change their mind and go to the aurors.
[this is also presumably what voldemort - undoubtedly at snape's request - tries to recruit lily to do.]
i have never believed that snape was taken on as a death eater in the expectation that he'd perform a combat role - there is a clear implication throughout the series that the only person he ever directly kills is dumbledore, and that he gets along badly with death eaters [such as bellatrix] who did take more violent roles in voldemort's terrorism.
so i presume that, when he leaves school, he ends up working as a personal brewer for voldemort - on a stipend presumably paid, at the dark lord's request, by either lucius or abraxas malfoy. i also presume that, outside of work voldemort specifically requests, he's given free rein to brew for other clients, study, experiment, and publish as he wishes.
and i further presume that if he trains with anyone, then that person is voldemort himself.
voldemort claims, in goblet of fire, to be interested in experimenting with potions. he appears to invent the potion made from nagini's venom which sustains his half-body prior to his resurrection - and i think the implication of the text is that he also invents the potion guarding the locket-horcrux. voldemort also evidently encourages snape's interest in the dark arts, and he also appears to have some influence over snape's comportment - the teen snape we see in order of the phoenix is extremely rough around the edges, in a way the adult snape, who both speaks and moves in canon very similarly to the adult voldemort, isn't.
voldemort taking such an interest in snape would - obviously - largely be a grooming tactic. snape clearly becomes a death eater because the organisation offers him a chance to belong and succeed which his class-background would ordinarily make impossible for him within wizarding society, and voldemort must therefore massively indulge his belief that he's never given the respect he deserves for his intellect. voldemort's obvious contempt for slughorn - who matters so little to him that he doesn't even bother to kill him - would, i imagine, also win snape round.
and by training snape in an academic rather than a combat sense, voldemort gains a valuable tool - someone he can place at hogwarts as a teacher to spy on dumbledore.
we can assume that voldemort was having dumbledore tailed throughout the first war - and, indeed, that this is what snape is doing when he overhears the prophecy - but that he couldn't watch him at all times because he didn't have a spy among the hogwarts faculty.
it is clearly voldemort who tells snape to apply for a teaching job in early 1980. he must also tell him to apply for the defence against the dark arts post [which we know snape canonically applied for first] - which means he must expect to be imminently victorious in the first war, since snape would only be able to stay in the position for a year...
the prophecy, which snape hears c. january 1980, obviously derails this belief slightly... and snape famously does not get the defence against the dark arts job for the 1980-1981 academic year.
how do we know this? because he tells us in order of the phoenix that he's been teaching at hogwarts for fourteen years. he says this right at the beginning of the autumn term in 1995 - so he clearly means that he's been teaching for fourteen previous academic years and the 1995-1996 year is his fifteenth. so... he started teaching at hogwarts in the 1981-1982 academic year.
voldemort settles on harry as the child the prophecy refers to after harry is born [so, after 31st july 1980]. we don't know how quickly he does this and we don't know exactly when snape defects to the order.
but, clearly, at some point during the 1980-1981 academic year, dumbledore hires snape to begin teaching from september 1981 onwards. he presumably tells snape to tell voldemort that his change of heart was because he didn't think snape was qualified to teach defence against the dark arts but that he does think he's qualified to teach potions [pointing, perhaps, to publications snape got out under voldemort's tutelage], and that slughorn's announcement that he intends to retire means that there's a position available. he then undoubtedly also tells snape to convince voldemort of the same pretence they'll use throughout the second war - that he's a loyal death eater passing information on dumbledore's movements to his master.
which is to say... when lily dies, snape has been in his job for at most nine weeks.
just imagine how miserable that must have been!
