Hi Emily,
You'd be far better served looking at songs from that period in the Bodleian Library Broadside Ballads Website and at the 17th century ballads on the UCSB English Broadside Ballad Archive (EBBA) website which includes the whole of the Roxburgh and Pepys Collections and many from the National Library of Scotland. These are all searchable. There were several printers in Southwark at an early date. Regarding existing traditional songs I have a pretty good grasp of what the corpus contains and none of the topics you mention ring a bell (as related to London). Bessy Bell and Mary Gray is a famous Child Ballad set in the era of the Plague but it's Scottish.

Having said that a vast quantity of songs originated in London as you would expect.