Hi Peter!
I am descended from the John Budd who came from Chichester, and settled in New Haven in 1637, after coming over on the "Hector". He later founded Southold, and Rye New York. They say Rye was named for the Rye in England, where the family resided for some time.
I am planning a trip to England next year, and would love to find out all I can about the English Budd's. Any info would be greatly appreciated, and I will gladly share all I know.