Strangely enough, I can fill-in some of the gaps in the map's history.
Jamie told me after the swap that he'd knicked it from Harry when he was in school and still had it in 2028. He thinks his dad and Hermione would have put it in the vault we're supposed to be looking for. The most recent generations of Potters are rather sentimentally attached to it, it seems.
Harry had the map when we were in school. I never realized what it was at the time, but he was using it fifth year when the D.A. was trying to hide from the Inquisitorial Squad. From Hermione's memories, I had the impression he got it a year or two earlier from the Weasley Twins. Something about liberating it from Filch?