In many of the Cooke family histories, including this one, Robert Cooke (1392 - 1423) is titled "Lord Mayor London" However in the lists of Lord Mayors online, I cannot see any Cooke name - only his son Sir Thomas - was mayor in the 15th C. Is this misinformation being passed down? Or has someone got some evidence that I haven't seen?
Kathleen, I can't link to it right now from my phone, but there's a PDF I uploaded to the Lord Mayors of London project that is the definitive list maintained by the office of the Lord Mayor. If your ancestor's name isn't on there, then there's no documentation of him having held the office. That doesn't mean it's completely impossible, since perhaps the record was lost, but it's exceptionally unlikely.
If his son held the title of Lord Mayor, it's likely that he also had some type of title; that's the norm, as I've learned through that project. So you might try tracking down peerage bios of the son and then seeing what they have to say about the father.
Also, I'm not sure why he's in the project if he's not on that list. I or someone else may have added him in error while we were first trying to gather up profiles. I'll look into it in the morning for you. :)