https://en.wikipedia.org/wiki/Before_Present
Apparently, the "before present" time scale is commonly used in archaeology and geology, and uses an epoch date of Jan 1, 1950. And there're two reasons for this: radiocarbon dating was first standardized in the 1950s, using a reference sample from 1950; and after the 1950s, widespread nuclear testing made radiocarbon dating inaccurate.
It's actually less arbitrary and shortsighted than it seems. Future civilizations are going to face a discontinuity in the archaeological record, and it'll be roughly at the Before Present epoch date.
Nuclear bombs put energy and radioactive particulates into the atmosphere. The effects of the radioactive particulates are fairly well understood. The amount of energy from even the largest bombs is negligible at those scales.
https://en.wikipedia.org/wiki/Modernity
See also "modern humans".