I'm not sure what the title of this post has to do with the request, but nonetheless this seemed interesting as the children seemed to disappear!
I turned to the 1921 census (for which, due to copyright issues, I cannot provide a transcription) and at first I struggled to find the family. After a while, I found them in Leeds, with Sarah enumerated under "Gilchrest."
She is not with her husband, but a man named Herbert Chadwick, for whom she is ostensibly a "housekeeper." Living with them are several children, namely: Evelyn Chadwick (born 1912),
Leonard Chadwick (born 1917) ,
John Chadwick (born 1920).
In 1921, James is in Hunslet, together with his daughter, Lillian (born 1913) and various other relatives. He is down as "married", but Sarah is not present, living instead with Herbert.
I am going to infer that Sarah and James have separated, and Sarah has started a relationship with Herbert.
In 1939, Sarah is still with this Herbert, with Leonard and Dennis also present. This time, Leonard is enumerated under Gilchrist.
If you search for the children's birth registrations, it seems that they were registered under both Gilchrist
and Chadwick. However, for some reason, I can only find the Chadwick registrations on the GRO, but I can see the Gilchrist registrations on FindMyPast, with the same volume and page number.
CHADWICK, LEONARD WHOMACK
GRO Reference: 1917 D Quarter in LEEDS Volume 09B Page 333
CHADWICK, JOHN WHOMACK
GRO Reference: 1920 S Quarter in LEEDS Volume 09B Page 479
CHADWICK, DENNIS WHOMACK
GRO Reference: 1922 D Quarter in LEEDS Volume 09B Page 398
CHADWICK, MARJORIE WHOMACK
GRO Reference: 1924 S Quarter in LEEDS Volume 09B Page 558
There is also another son born in 1927.
Lillian, however, is only under Gilchrist:
GILCHRIST, LILIAN WHOMACK
GRO Reference: 1913 D Quarter in HUNSLET Volume 09B Page 790
It seems Lilian is James' child, whilst the others are Herbert's.
Queenie

EDIT: The reason they are difficult to trace may be because they are switching between surnames.