Your question is based on the incorrect assumption that this French sentence contains a double negative while it doesn't.
There is only a single negation "aucun xxx ne xxx" and no negation but a restriction in "ne xxx que". A restriction is limiting a magnitude, it doesn't reverse anything.
A real double negative might have been:
Aucun ne parlait même pas un simple mot de mandarin.
but this sentence is confusing and incorrect in French too.
You also dropped several ellipses present in the original sentence which reads:
Ludovic : Ah oui, je me souviens… je me souviens aussi des cours de chinois, j’ai un petit… j’ai notamment un souvenir… donc pareil, on avait un… un journal de bord. Ou on était peut-être plus… enfin dans mes souvenirs, on était plus focalisés par exemple sur ce que nous on ressentait mais après… enfin, on regardait aussi un petit peu la technique de… de l’enseignante puisque donc tout… le cours en entier se faisait en chinois, enfin en mandarin.
Jessica : Ouais.
Ludovic : Et le… le problème c’est que… bah forcément aucun de nous ne parlait un… ne serait-ce qu’un mot de… de mandarin.
Jessica : Hm hm.
So the originally intended sentence was:
aucun de nous ne parlait un mot de mandarin
i.e. none of us spoke one word of Mandarin.
but in the middle of the sentence, the stuttering speaker replaced it by:
aucun de nous ne parlait ne serait-ce qu'un mot de mandarin
Here, an intensifier idiom has been inserted into the sentence (it is an incise). It says ne serait-ce qu'un mot, i.e. should it be just one word / not even a single word so the full sentence might be translated to:
None of us spoke one… even a single word of… of Mandarin.
Another fact that might be confusing is that the first
ne is not per se a negative mark but part of the split negative
aucun ... ne. There is no extra negative here so
Aucun de nous ne parlait...
really means, as already stated:
None of us spoke...
and not
None of us didn't spoke...
Moreover, the second ne is part of fixed French formal adverbial phrase ne serais-ce que which technically just means even1 so as already stated, there is no double negative either (i.e. a combination of negations that lead to a logically positive sentence, like I don't know nothing about grammar), and even less a triple negative.
The sentence can then simply be rephrased as:
None of us spoke Mandarin, even a single word of it.
1gabrielwiler.com states: On pourrait aussi dire que « neseraiske » est un adverbe de perspective proche de même.