For pedants like MtheGM and myself only:

The earliest printing of "BA" seems not to have occurred till ca1690, or about twenty-five after Pepys heard it.

No conclusion may be drawn, but that may be a long time to wait for either a truly well-known anonymous song or a popular stage song to get into print. Did Mrs. Knipps write it herself? If word-of-mouth currency began about 1665, and was at first restricted to a relatively limited social circle, it might well have taken the ballad a long time to come to the attention of the printer.

Just thinking aloud.