Uncountable nouns will always take the singular, except when they're being thought of as multiple discrete units. For instance, magna pecunia = a vast sum of money, whereas magnae pecuniae = several vast sums of money. You could negate either statement.
The negation of a countable noun depends on the conception of the speaker. If the prior context or general context has the speaker thinking of the negated noun as a group, the negation is likely to be plural.
Nepos, Vitae, Atticus, 14:
nullos habuit hortos, nullam suburbanam
aut maritimam sumptuosam uillam
He had no gardens, no sumptuous suburban or maritime villa
In this context, Nepos is taking as his baseline expectation that a rich Roman would have a country estate with a bunch of gardens. That mental conception determines the form of the negation.
Cicero, Pro Flacco, 12.27
'nulli erant praedones.' quid? nullos fore quis praestare poterat?
"There were no pirates." What's that? Who could have certified
beforehand that there wouldn't be any?
In context, Cicero is defending Flaccus for having levied money to raise a fleet, ostensibly to deal with pirates. There ended up not really being any pirates, mostly because Pompey had done such a thorough job previously clearing them out. Since the concept of pirates was introduced in the plural, the negation also naturally takes place in the plural.
Cicero, Orator, 100:
eloquentia ipsa quam nullis nisi mentis oculis videre possumus
eloquence itself, which we cannot see with any eyes save the eyes of
One normally conceives of seeing with two eyes, so the plural negative is to be expected.
Now, sometimes we might get a negative in the singular even where the positive statement would normally be plural. For instance, de aliqua re verba facere is to speak on some topic. However, the negative idiom takes the singular, nullum verbum facere. This seems to carry an intensified meaning, "to say not a single word."
Lastly, it may be rhetorically desirable to alternate between singular and plural:
Cicero, Post reditum 20:
pro statu civitatis nullas sibi inimicitias, nullam vim, nullos
impetus, nullum vitae discrimen vitandum umquam putavit
in defense of the state, he thought that he should not ever shun any
personal enmities, any violence [against him], any attacks, any danger
to his life
Cicero represents Publius Sextius as having made multiple enemies and endured multiple attacks, none of which he shunned.
In reference to the video you linked, my sense is that nulla bracchia would be the normal negation, since arms are usually conceived of in the plural, but nullum bracchium is still grammatically correct. It would be more marked: "not even a single arm."