I've been to the Library today and looked at Ancestry and Find My Past.
Regarding possible parents of John Fleetwood son of a William and Elizabeth, baptised in 1700, the closest I could find to Fleetwood was this Clandestine marriage, 24 of July 1692:
"William Flethed (or Flethod), and Elizabeth Roberts."
The original image looks like "Flethed/Flethod" to me.
I could not find any people with baptisms of this surname. So possibly could be a misspelling of "Fleetwood"?
Regarding possible parents of the John Fleetwood baptised in 1729, at Wapping, near Stepney, I came across this Clandestine marriage in 1725:
"(Joh?)n Fleetwood of St. George's, and Mary Symson."
You can only see the last letter of the Fleetwood's Christian name unless you zoom right into the original image. And there definitely looks to me like there are three letters before the "n", the last of which looks like a "h" in the style of other "h's" on the same page.
Just as an aside, there was also a:
"John Fleetwood of Mile End New Town, a Weaver, and Jane Grocer of same, spinster." They were married 24th of April 1698 at St Dunstan, Stepney.
I didn't have much luck looking for burials of John Fleetwood:
The 1763 burial was at St. Andrews Undershaft. He was: "a child of Kensington."
I found one in 1780, but it just stated the name and no age. I forget where in London this was.
There was one buried Dec 29 1788. It just said: "of Wilcot Place."
A John Cook Fleetwood was buried Jul 24 1789, at St. Paul, Shadwell.