Here's how I think that works.
A variant, let's say Joe creates a nexus event, which splits the timeline. So there is, for now, a Prime Joe and a Variant Joe. The TVA goes into the split timeline, snatches Variant Joe, and then blows up that timeline with their little jars, erasing it completely. Variant Joe is taken to the TVA, where he is either disintegrated for not taking a number or some other offense OR he is "reset" and has his memory wiped and becomes part of the TVA.
But Prime Joe continues on, unaware that this ever happened.