Have you got her first marriage certificate for more information about her father?
Then her birth certificate for her Mother, and the marriage certificate of her parents to help you track James.
Why do you think he may have married several times?
If you find Margaret in 1939, it should have her final surname on there.
Is this Arthur P Dyson?
Deaths September quarter 1954
Arthur P Dyson 72 Wakefield 2d 524
Births June quarter 1882
Arthur Percival Dyson Barnsley 9c 204
Added:
There is a Percival A Dyson born 1882 living in Morley in 1939 - worth investigating the household?