Have been looking for, and not finding, incontrovertible proof that she is indeed the daughter of Nicholas West, of Burton, and his first wife (here given as Anne Benedict). There is a lot of Wild Mass Guessing http://tvtropes.org/pmwiki/pmwiki.php/Main/WildMassGuessing that the "Joane West" who was twin to a John West, and child of Nicholas and Anne West, is the same person as the Joane (no last name) who married Robert White. But there's a joker in the deck....
Nicholas White's son by his first wife, Richard West Richard West, married his stepsister Magdaline Staple Maudlin West, daughter of Nicholas West's second wife Philippa and Philippa's first husband. And they had a daughter Joan Joane White who most certainly did marry John White John White of Lancaster , son of Robert White and *his* wife Joane.
That would make John and Joan first cousins with some extra twistiness. They might be working with a small gene pool, but - *that* small?