Hi, my Weekes/Weeks family originated in the Bexley, Kent, England area. Half the family converted to Mormonism, and came to America. The other half (with the widowed mother who remarried) went to Australia with the new husband's family.
There is a possibility that one of the members of the family went to Barbados instead. Do you have any family history that goes back 4-5 generations? There are other Weekes/Weeks families in the town I live in that I have not been able to find any connection, but that doesn't mean there isn't.
You can contact me at firstname.lastname@example.org
if you wish.