What's the soonest life could have evolved after the Big Bang?

That depends on what you mean by ‘life’, but it’s reasonable to say that the absolute minimum requirement is the existence of molecules made from relatively heavy elements like carbon, oxygen and nitrogen. In that case, we can certainly exclude the existence of life for several hundred million years after the Big Bang. That’s how long it took for the appearance of the first stars, the thermonuclear reactions of which are needed to ‘cook’ the hydrogen and helium formed in the Big Bang into heavier elements. These first-generation stars are believed to have been far more massive than our Sun, and to have burned brighter but for a much shorter period of time – perhaps just a few million years before exploding. So that puts the earliest possible date for the emergence of life at around several hundred million years after the Big Bang.