Pamela shared a comment with me:
“The more I read the more important they are becoming; they owned lands all over the west Midlands far into the past; another very interesting connection is that when Thomas Andrews died in 1564, Robert Dudley, Earl Leicester was granted Charwelton (one of many, many grants from Elizabeth I on his being named Earl of Leicester that year). Dudley had reorganized land ownership, but basically he just received income from the estate, the Andrews were still on it. I never realized that Dudley was a Puritan, which makes an even more interesting backdrop.”
There's definitely something "off" abut William. He can't be the son of Sir Thomas Andrew of Ilmington, because the dates don't work. He might be the son of Thomas Andrew, Sheriff of Northamptonshire (son of Sir Thomas and first wife Katherine Cave - who was a direct descendant of Sir Thomas Malory). Or he might be out of place altogether and the product of someone's wishful thinking.
As to Leicester, while he clearly had strong Puritan sympathies, he never made a final break with the Church of England, and continued to have and support Catholic friends and allies also. Seems his religious views were moderate in all things.