I think you might be a little optimistic here! I'm an English Chew (b1949). My family comes from Blackburn in Lancashire(you need a map) where it is a fairly common name. I now live in Hyde in the foothills of the Pennines (hills running north to south in the middle of northern England - don't worry about the geography) Around here, running over towards Oldham and points north there is the Chew Valley. I have always understood that the Lancashire/Cheshire/Yorkshire Chews originated in and took their name from this area.
There is also a Chew valley in Somerset and I have a friend who lives at Chew Magna near Bristol. (We are talking south-west England here).
Many years ago in an old book on English surnames I found the theory that the name was a corruption of "Chough" (pronounced "Chuff") which is a bird of the european blackbird family. I felt then and still do that this is a somewhat laboured idea. Surnames as we know them now didn't become very widespread until the Middle Ages in Britain. They generally descibed a person by their personal characteristics eg John White (a blond- or white-haired person) or John Armstrong, or their job eg John Fletcher, the arrow maker, John Smith, John Miller, etc., to distinguish them from others of the same christian name. Likewise Robin-son, John-son etc. As people began to move around, after the Black Death in the 14th century effectively put an end to the village feudal system, their new neighbours would give them the name of the place of their origin eg John Chew, John Hyde, etc., to distinguish them from the locals. The idea of calling someone after a bird seems a bit odd unless he was a great whistler!
The French connection is quite beguiling but it's not one that features in the family folk-lore in this branch of the Chews.
I've been to Battle Abbey where the Battle of Hastings took place and I can assure you that if everyone alleged to have fought for William the Conqueror and been rewarded by him had actually been there the battlefield would have been so crowded that there would have been no room for the Saxons!
Neither is there any suggestion in the family that we have come down in the world and I've never come across any Chews among the British aristocracy - major or minor!
That is not to say, however, that some of the Chews who high-tailed it off to America, whether voluntarily or under duress, didn't "acquire" noble connections on their arrival. I under-stand that it was quite common for new arrivals to reinvent themselves.
I hope this doesn't disappoint you too much. If you find evidence to the contrary I'd be very interested to hear about it.
Incidentally, am I right in thinking that there was a Robert(?) Chew who was Secretary of State at the time of (your) civil war? And I'm sure there used to be a Sam Chew who was involved in the technical side of film or TV programme making. I remember seeing the credits.
\good luck in the search.