I've had a chance to dig out sources for William Bayley and updated his profile (Bayley-42): there's no evidence that he was a knight, but there is a lot of evidence from his will about his extended family that may help someone take the family back a few more generations. Some of the other families mentioned do have longer pedigrees if William (or his wives) can be linked into them.
The link through to Mary seems strong: William left an only child called Mary in 1602, in 1605 Mary Bayley married Roger Fowke, and in 1615 a son of Roger Fowke of the Lea is baptised.
Given they're project protected, I've only added some new sources to Mary and Roger's profiles. They need some more work from someone within the project:
The references to William Bayley being a knight should be removed from Mary and Roger's profiles. The contemporary records refer to him as Mr William Bayley.
Mary's birthplace is almost certainly wrong. I can't find a baptism for her in Staffordshire or in London. It looks like someone has mixed her up with a Mary Bailey baptised in Goudhurst in Kent in 1590, but they're not the same person. I suggest in the absence of evidence her birthdate is left as 1590 (for the reasons set out in the profile) but birthplace changed to England.
Roger's parentage needs a look, as there are some discrepancies between what's in Richardson and the Visitations and pedigrees (the dates are out by a generation or two)