Just had a look at the 1921 census on Findmypast - just by searching and hovering the cursor over the details in the results.
Arthur & Lucy PRIESTLEY with Clarence A. PRICE are together in a household in Wakefield, with three other people. One of them is a Margaret ROBINSON, born 1920 in Bradford. Now she must be the Margaret PRIESTLEY in the 1939 Register.
I looked for her birth in the GRO online index. The only Margaret ROBINSON that came up as born in June Quarter 1920 in Bradford was:
June 1920 Margaret ROBINSON, Mother's M.S. -- Bradford 9b 323 Occasional Copy A
Occasional Copy A usually refers to some alteration having been made to the name later on.
When I looked up this reference on FreeBMD, there were two entries for what was clearly the same child:
June 1920 Margaret ROBINSON, Mother's M.S. SYKES Bradford 9b 323
June 1920 Margaret SYKES, Mother's M.S. SYKES Bradford 9b 323
There is no birth registration for a Margaret SYKES in the GRO online index.
So my assumption is that Margaret was the illegitimate daughter of a Mrs. ROBINSON formerly SYKES.
I am not aware of any family connection to SYKES or ROBINSON in either Lucy's or Arthur PRIESTLEY's family.
There is also an E. George GUDGIN, born 1899 in Nottingham, in the household in 1921, as well as an Edith GUDGIN, born 1895 in Retford, Notts.
He was George Edward GUDGIN, born June Quarter 1899 in Nottingham District, Mother's M.S. ARCH.
I could not find Edith's birth as GUDGIN, so I suppose she was George Edward's wife - but I can't find a marriage for them either.
Just thought it may be of interest to someone.