That last common ancestor was alive in the early Cretaceous, at least 130 million years ago. It is not clear now much further back down the bird evolutionary tree these inefficient penises might be found, but it is at least possible that the trait first appeared in the dinosaurs from which birds evolved.
Gregory Erickson of Florida State University in Tallahassee says it makes sense that dinosaurs had penises, but so far the fossil record has remained silent on the matter. It would take a dinosaur fossil with exceptional soft tissue presentation to establish whether dinosaur sex was as brief as the birds' or as leisurely as mammal and reptile couplings.
A dinosaur penis would be far from the oldest in the fossil record, though: that crown is currently held by a tiny crustacean found in rocks 425 million years old.