Many thanks for all the contributions. It looks like this is remaining a tough nut to crack, but I'm grateful for some possible leads.
If the 1911 Hammersmith Jane Cox is not mine, I must admit that I have not found a more likely candidate in previous censuses.
Millipede points out that this Jane says she had no children (I think I assumed she was saying she had no child alive), which ties in with ShaunJ’s suggestion that Alice may have been a niece (or adopted?).
Jane had two sisters. Mary Ann Miller (born Stoke Poges 1859) married Walter James Spencer at Acton in November 1883, and her 10 children (per 1911) are accounted for..
Elizabeth Miller (born Stoke Poges 1861) had two illegitimate children before she married George Kreiner on 26 December 1888. Elizabeth Rose Miller was born April 1885 at Queen Charlotte's Hospital, Marylebone, and George Miller was born Dec 1886 at St Pancras Workhouse. Elizabeth Rose lived with a great aunt, and George was brought up as a Kreiner.
According to Elizabeth Rose's birth certificate her mother was living with the Coxes at 87 Westbourne Street !
Given the birthdates of these two children I don't think Elizabeth can be Alice's mother, but I will have another look.
I will certainly work on the four Alice Cox births that ShaunJ suggested, to see if there is any connection to a brother of Richard Cox.
I was looking for the Napier-Cox couple, suggested by Jolee, in the 1911 census (without much luck!), but I think Ladyhawk has found this was not my Alice Cox.
ShaunJ – however did you find that 1909 burial for Richard Cox? Can I ask where you saw the admissions register?
Millipede suggested emigration. Nothing has leapt out at me, but I admit I'm not an expert when it comes to searching the lists. Certainly a lead to follow up – after all, family legend says we own half of Saskatchewan, but I've never found a likely candidate !