Re: Any British Perhams?
You say your father was born 1899 in Wandsworth. And you say his father, Joe (Joseph?) was born about 1820 in London. This means that Joe, the father, was about 79 years old when your father was born.
Is that right? Or are there a couple of missing generations?
The only Joseph Perhams on the 1881 census are the following father and son:
1881 Dwelling: Welshs Green
Census Place: Wells St Cuthbert, Somerset, England
Source: FHL Film 1341580PRO Ref RG11Piece 2409Folio 62Page 3
Marr Age Sex Birthplace
Joseph PERHAM M 66 M Thorngrove, Somerset, England
Louisa PERHAM M 66 F Thorngrove, Somerset, England
Joseph PERHAM U 28 M Mendip, Somerset, England
As to Domesday Book. Mention in this source does not mean they were Saxons, as most of the landholders in DB were Normans, who had taken the Saxon lands. I do not recall seeing any Perham in Domesday book.
The earliest Perham of note that I have come across, was an MP in East Anglia in the 15th century. (Norfolk, from memory, though I could be mistaken.)
Most Perhams however seem to originate from the West Country, though I have made no detailed study of this.