I’d say:

Personne n’a fait d’exercice. (Nobody has done any exercise.)

To give you the long answer:

In fact, you were not trying to make a double negation, but to build a negative in French.

You thought that "aucun" was a negative pronoun, meaning "no-one". You were not far, but you fell into a treacherous trap: it is a positive. "Aucun" really means "a few" (or rather meant, because it is quite obsolete) and it is a plural:

D’aucuns sont venus.

The only negation in French is ne (n’).

Aucun… ne => No a few => No one

Since in French (as a thumb rule) zero is a singular, you now have to say:

Aucun n’est venu.

But "aucun", used in that negative way, should refer to people/things you already referred to (why that is, is probably a question of habit). So if it is clear which group of students you are talking about, it is perfectly correct.

To avoid that problem, you should use "personne". Personne is also a positive. It really means… "someone".

"As-tu vu personne?" = "As-tu vu quelqu’un"? => "Have you seen someone?"

In French, the only negation is "ne" (n’).

ne… personne / personne… ne => "not one" => nobody

Hence

Personne n’a fait…

The problem is, modern French usage tends to forget this and the meaning of "personne" is reversed, to mean "no-one":

– As tu vu quelqu’un?

– Non, personne

The only way to tell is with the context. I tend to avoid this, if I can, because it is ambiguous.

"Un/de" in negative sentences

And finally, a little detail (not mandatory, but it will improve the quality of your French): in a negative sentence, "un" tends to become "de".

– As-tu vu un chien?

– Non je n’ai pas vu de chien.

Hence the final answer could be:

Personne n’a fait d’exercice.

The logical, negative answer to the question:

Chacun a fait un exercice ?

can be:

Non, chacun n’a pas fait un exercice.

or the equivalent:

Non, tous n’ont pas fait un exercice.

meaning at least one person didn’t do one exercise¹.

Non, il y en a au moins un qui n’a pas fait d’exercice.

The reply suggested elsewhere so far:

Personne n’a fait d’exercice.

is negatively answering to a different question which is:

Quelqu’un a fait un exercice ?

^{1 – or everyone did more than one exercise but this would be controversial.}