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If a sentence begins with a negative or semi-negative word / phrase, it causes subject-operator inversion :
"In no other way can the matter be explained."
"Hardly had I seen Sam when he started walking."
Such inversion isn't found in the following sentence :
"In no time we cleared the table." (NOT, In no time did we clear the table.)
Could you please clarify it?
There is an article from Lund University in Sweden that explains:
The inverted order only occurs when the whole clause is affected by the initial negation/restriction. Thus, there are cases where what appears to be an initial negative element does not trigger inverted word order.
In no time at all, the two doctors had more patients than they could handle.
Here, the meaning of the sentence is affirmative rather than negative and therefore the clause retains its normal S-V order.
Your example sentence is also an affirmative one (the table was cleared). As I pointed out in my comment to your question, "in no time" is actually an expression that means "very quickly", so the sentence can be rephrased as:
We cleared the table very quickly.
You can see more examples of where inversion is blocked even in the presence of negative constructions in the Wikipedia article on negative inversion.
