Do flat warts, both old and new clear at the same time that the immune system recognizes the virus?

Sometimes: Such warts are so superficial in their skin location away from blood vessels/capillaries that they do not trigger much of an immune response. Over a period of several years they may shed after immune activation.

Yes but: “Recognition” is very problematic and may or may not happen to YOU (more likely not). Sometimes warts (new and old) simply disappear- that’s when the immune system “recognizes” them and makes them disappear. Mostly, however, they need to be burned off, chemically or by laser in order to cure