Through an ancestry site I traced our ancestors to John Gay, the poet laureate of Great Britain. His fame was largely due to the immensely popular "Beggar's Opera" that later became the "Three Penny Opera." He was also Alexander Pope's best friend. This branch of the Gay family backed the King when Cromwell raised his Roundhead army. When it became clear the King was going to lose, the family fled to France.
They lived in France for another generation, but with typical English disdain of the French, they left France for Switzerland. There were English and German/Swiss wives along the way and at least one French grandmother in the woodpile. Fortunately that bloodline is well diluted by now.
My grandfater John Gay came from Interlaken Switzerland near the French border, but when alive, he recounted his lineage of the English branch to his children. If you are interested, he said many of the Gay families left London, Lincolnshire and Leicester during the rule of Cromwell and ended up in Switzerland. That may be another source to find your English ancestors.