I'm with Sandra on this. I don't think there was a divorce.
Lydia's husband, Patrick Giltnane clearly states on the 1911 census that he is married, obviously still to Lydia as indicated by the length of the marriage.
Also on the 1911 census William Langford states that he is widowed, (while Lydia states she is married).
If you look at the children from both marriages you can see that Ermengard Langsford was born Dec 1901 and must have been a child of William and his wife Hope.
Lydia must have been living with Patrick until around December 1904 (when their youngest child, Cecil was born).
William's wife died in Jun 1905 and it looks as though Lydia left her husband and children between then and the birth of Doris in 1908.
I've double-checked the records and there's no trace of a marriage between Lydia and William