I don’t think boules de neige has to be taken as a metaphor here. Mme Walace is talking about kids when “they’re cute”, in her mind it qualifies the time before it gets into their heads to get into any sort of mischief. After that time kids (she’s still talking about kids), start throwing snowballs. In other words Mme Walace could have said:

Au début, c’est mignon jusqu’à ce qu’ils découvrent les boules de neige… (/et après ils commencent à lancer des boules de neige).

I don’t agree with your translation of et après, c’est les boules de neige. It should be “and then, there are snowballs / and then we get the snowballs”. The question on why the English subtitles mention chestnuts is not really a question for French Language¹, but I’ll still tell you what I think.

Here “chestnut” designates the fruit of the horse-chesnut, i.e. the non edible variety, commonly known in Britain as “conkers”.
In Britain (I have no experience of living in an English speaking country other than Britain so I can’t tell if that applies to North America as well) there’s a tradition of throwing horse-chestnuts (and playing conkers) in the autumn, just as there’s a tradition of throwing snowballs in the winter. Adding chestnuts (or conkers as they’re known in Britain) is telling something to an English speaking audience in the way of kids’ mischievousness. In France, kids might occasionally throw conkers around when the time comes but never at such an organised scale as they do in Britain. To a Br. English audience at least, snowballs and conkers will go together as a sometimes dangerous activity mischievous kids will play.
I’ve dug out a ten-year old Guardian article that says: “A minority of schools have banned conkers and snowball-throwing amid concerns that injuries could expose schools to the threat of legal action.”²

Boule de neige can be used figuratively in the phrase faire boule de neige but it is not relevant in this Amélie Poulain extract.

¹ _{They might not leave it open on ELU either.}
² _{Actually the article is about ministers calling on risk-averse parents to stop “wrapping their children in cotton wool”. Government’s advice to parents: make sure your child plays conkers}

In French*, “boule de neige” (snowball) represents something that will have a kind of exponential effect, just like a snowball running down a hill and accumulating more and more snow as it goes down.

We often use the phrase “un effet boule de neige” which translates to “a snowball (exponential) effect”.

(*) I mean “French of France”. It may be different in other “French-speaking” countries