Learn Russian: Если бы не оперативность коллег, мы бы не успели. - If it weren't for the promptness of the colleagues, we would not have made it in time.

This is the specific rule for the Если бы не (If it weren't for) construction. In standard negations, you would often use the Genitive case, but this exact phrase is a fixed idiom that is always followed by the Nominative case.

Is коллег in the Genitive plural because of the negative не?

No, it's in the Genitive plural because it describes possession (the promptness of the colleagues). Also, notice the "zero ending": because the singular коллега ends in -а, the Genitive plural simply drops that vowel to become коллег.

The translation says "would not have made it," but успели is just the standard past tense. How do we know it's a past hypothetical?

In Russian, conditional sentences use the particle бы paired with the past tense to express any counterfactual idea, whether it's present ("would not make it") or past ("would not have made it"). You have to rely entirely on the surrounding context to know the timeframe.

I see the particle бы twice in this sentence. Do we always need to repeat it?

Yes. In a full hypothetical condition, you need бы in both the condition clause (here, as part of Если бы не) and the result clause (мы бы не успели).