Here is some information that may be helpful.There is much that will probably never be known for certain about Bazemores before 1700.
The John Bazemore (Baismore)from whom we all seem to be descended paid taxes in 1730 in Norfolk County, VA.He and his mother, Elizabeth Bazemore (Bazmore), paid taxes in Norfolk County, VA, in 1731.
John Bazemore (Basemore) was administrator of the will of Mathew Edwards in 1734 in Bertie County, NC.
John Bazemore (Bassmore) proved his rights to land in Bertie County, NC, in 1742 listing himself, Mary (his wife), Jesse, Thomas, John and Sarah.
Elizabeth Bazemore (Baysmore), John's mother, petitioned to be relieved of poll taxes as a very aged and infirm person in 1761 in Bertie County, NC.
Thomas Bazemore (Bozemore), John's father, was a witness to the will of Thomas Wooden in 1712 in Norfolk County, VA.
The records of County Surrey, England, show a Thomas Bazemore (Basmor) christened September 18, 1681 at Kingston on Thames.His father was John and mother not listed.Four other children were christened earlier, apparently to this same John.They were Elizabeth (Basmoore)in 1671, Ann (Basmoor)in 1674, James (Basemor)in 1676, and a male child (Basmer)in 1678.
Earlier, a John, wife not listed, had four children christened at Leatherhead.They were Ann (Basemore) in 1660, John (Basemore) in 1663, Mary (Basemore)in 1666, and Katherine (Basemore) in 1668.
The records for County Surrey are obviously not complete, but this Thomas is the only one for whom a christening is recorded. A Thomas, obviously not the same one, had a child, Ann (Baismore)christened in 1681 at Carshalton.A Thomas, wife Anne, had a child, Sarah (Basemore)christened in 1696 at Epsom.
I have seen a birth date of 1665 for Thomas, but have not seen a record to support it.
In the records of Barbados, there is a mention in 1692 of a John Bademore, son of Thomas Bademore, in the will of John Hitchcock.This is not necessarily a reference to a Bazemore.
There was a marriage of Ann Basemore to John Kingstone in 1698 in Barbados.This is probably a Bazemore, but who?
Henry Baismore witnessed the will of Richard Settle in Barbados in 1671.This may be a child of Nicholas who had a child, Henrieus (Baysmore) christened in 1634 at Ewell in County, Surrey, England.
There was a John Bazor, son of Thomas and Elizabeth Bazor, christened in 1689 in Barbados.This is probably not a Bazemore, as Bazor is an established surname and not a variation of Bazemore found anywhere else.
While there are several variations of the spelling in the County Surrey records (even for what appear to be children of the same family), the Bazemore spelling appears several times.Even in this country, several variations can be fount in census records, marriages, wills, land records, etc., but the Bazemore spelling seems to have survived among known descendants.
I am sure this does not provided hoped for answers, but maybe it will explain some of the conflicting versions in existence.