It can be done, and in the hands of a good composer, it could be done well.
A Trivial Example: is it really modulation?
Using the loosest definition of "modulation", this round moves between C major and G major (or, at least, between two pitch centers), but since the moves are "in sync" (each has two measures of C followed by two measures of G), there's no problem.
Of course, in addition to the questionable claim of modulation, there's also little semblance to staying within the rules of tonality (e.g., parallel fifths).
Another Trivial Example: Bi-tonality
Here the modulation isn't debatable, but the aesthetics might be. For the sake of discussion, we all love bitonal music, so this round presents no problem for us.
Another Trivial Example: "Hocket"
It is assumed without proof that we could create a round in which one part is always resting while the other is singing.
A Final Trivial Example: Serialism
It is also assumed we could create a similarly trivial example via serialism.
Less Trivially, Yes.
A Less Trivial Example: Tonality with "real" modulation
A better definition of modulation is there there should be a cadence in the new key, so let's at least require a leading-tone-to-tonic motion. Also, let's stay with the spirit of the question, which seems to presume familiar tonality.
This is by no means a "good" round, but it at least demonstrates the possibility of modulation — again, if only on the fairly trivial side of things. Here, as with the first example, the modulations are "in sync", so there's no concern about being in two different keys simultaneously.
Another Less Trivial Example: Bi-modality
Again, the spirit of the question seems to presume different tonal centers, but there are definitions of "modulation" that admit shifting between major and minor within the same key, as the below example does.
Another Less Trivial Example: Relative Keys
Let us assume without proof that we could create a round that alternates non-trivially between C major and A minor, for example.
But Non-trivially ...?
The above examples support a sort of proof-of-concept idea of rounds in two keys simultaneously. But, as stated, the spirit of the question seems to be
Can there be an "interesting" round that 1) follows the rules of tonality, 2) doesn't delve too deeply into chromaticism, and 3) has parts simultaneously and unambiguously in two different key signatures.
Yes. Here is an example of how this might work. I've chosen the keys of C major and A major, because they share (enharmonically) a leading-tone diminished chord. This allows for simultaneous modulation in both keys (and even opens the door to a four-key round).
disclaimer: I am no composer.