I might have a lead for you. My dad did some research at the Society of Genealogist's Library about 15 yrs ago and found this in Boyd's Citizens of London.
Thomas Jacks - St. Dunstan's in the East Parish Citizen and Clothworker Rent - 20 pounds Married - Feb 16, 1624 to Anne Atkins Children - Job 1626 Jane 1628 John (Jone) ? Died - July 26, 1642
This might tie in to references of a John Jakes, Jone Jacks, or John Jack that dad found in the Baltimore area, having emmigrated from England in the mid-1600's.
Dad also thought that the connection made sense, considering the first two generations of Jacks in the US had sons named Thomas.
So maybe, Jone(John) Jacks, son of Thomas Jacks of London was the John Jacks, immigrant to Baltimore and was the father of Thomas Jacks(b 1675). Unfortunately dad wasn't able to find better documentation, and dad died in 1997, so I can't ask him.
Hope It helps, I'm from the Missouri Jacks if you want to see where we connect.
I am Aaron Andrew  Jacks (Darrell Glenn , Grundy , David Gregg , James Dee , Sidney C , Thomas , Richard , John , Thomas , Thomas )